Chain rule derivatives examples with solutions

Chain rule derivatives examples with solutions. And finally multiplies the result of the first chain rule application to the result of the second chain rule application. Solution This is an application of the chain rule together with our knowledge of the derivative of ex. Unit 6 Integrals. There is a formula we can use to differentiate a quotient - it is called thequotientrule. When you use the partial derivative, you treat all the variables, except the one you are differentiating with respect to, like a constant. 4x2 9 x2 16. Let f(x) = 6x + 3 and g(x) = − 2x + 5. Refer to the example of the chain rule to understand it. Example: Differentiate the function y = e sin x. Since the functions were linear, this example was trivial. 6. Chain Rule Example #1. Find the Derivative Using Chain Rule - d/dx. Example of chain rule: Consider a function: \(g(x)=\ln(\cos x)\). Hence, the Summary of the quotient rule. 8 Example Find the However, to find the derivative of a function using the chain rule, one must be aware of the basic differentiation formulas. In this section we’re going to show you an example of using chain rule. Calculus 1 8 units · 171 skills. Earlier in the class, wasn't there the distinction between The derivatives of parametric equations are found by deriving each equation with respect to t. We restate this rule in the following theorem. The chain rule applied to the function sin(x) and √ x5 −1 gives cos(√ x5 The Chain Rule formula is a formula for computing the derivative of the composition of two or more functions. According to the chain rule, h (x) = f (g(x))g (x) = f ( − 2x + 5)( − 2) = 6( − 2) = − 12. Power Rule of Differentiation. Their derivations are similar to those used in the examples above. Example: Find the derivative of x 5. Worked example: Derivative of ∜ (x³+4x²+7 The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. 13 Logarithmic Differentiation; 4 Jun 26, 2023 · At least initially in working particular examples requiring the chain rule, it can also be helpful to clearly identify the inner function \(g\) and outer function \(f\), compute their derivatives individually, and then put all of the pieces together to generate the derivative of the overall composite function. We can rewrite h ( x) = 2 cos 3. Summary of the product rule. Why is the chain rule called "chain rule". Such as sin(x^2), where one function is the sine operation and the other is the squared operation. Show Step-by-step Answer: The chain rule explains that the derivative of f (g (x)) is f' (g (x))⋅g' (x). Worked example: Derivative of sec (3π/2-x) using the chain rule. If y = b x , y = b x , then ln y = x ln b . Examples to show how to use the different derivative rules. Treat the \(x\) terms like normal. For example ∂/∂x [2xy + y^2] = 2y. Such an example is seen in 1st and 2nd year university mathematics. If we have two functions x(t) and y(t) then we calculate dy/dx as follows: First, we calculate the derivative of x(t) with respect to t. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule. Solution. Proving the chain rule for derivatives. Derivative Rules for Parametric Function. Such relationships between a function of time and its own derivative are examples of differential equations, a topic we revisit in Chapter 11. the derivative exist) then the quotient is differentiable and, ( f g)′ = f ′g −f g′ g2 ( f g) ′ = f ′ g − f g ′ g 2. All derivative rules apply when we differentiate trig functions. Unit 3 Derivatives: chain rule and other advanced topics. Let's see what happens when we try to compute the derivative of this function just using the definition of the derivative. Differentiate using the chain rule. 10 : Implicit Differentiation. Then differentiate the. . Examples Using the Chain Rule of Differentiation. If you’d like a pdf document containing the solutions the download tab above contains links to pdf’s containing the solutions for the full book, chapter and section. The Chain Rule is a little complicated, but it saves us the much more complicated algebra of multiplying something like this out. 4 Recognize the chain rule for a composition of three or more functions. At this time, I do not offer pdf’s for solutions to Dec 29, 2020 · Example 60: Using the Chain Rule. Learn. Search this book. d dx (e3x2)= deu dx where u =3x2 = deu du × du dx by the chain rule = eu × du dx = e3x2 × d dx (3x2) =6xe3x2. It is a rule that states that the derivative of a composition of at least two different types of functions is equal to the derivative of the outer function f(u) multiplied by the derivative of the inner function g(x), where u=g(x). The chain rule is a rule for differentiating compositions of functions. For example, the derivative of x^2 is 2x, which means at any point on the curve, y is growing at a rate of two times x. Nov 16, 2022 · The chain rule really tells us to differentiate the function as we usually would, except we need to add on a derivative of the inside function. Chain rule intro. For the sake of clarity g(f(x)) would be sin^2(x). We now present several examples of applications of the chain rule. This means that we can write as follows: Nov 17, 2020 · In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. Step 3: Now, determine the inner and A more general chain rule. In this unit we will state and use the quotient rule. 9 Lets compute the derivative of sin(p x5 1) for example. So what does the chain rule say? The chain rule is a rule for differentiating compositions of functions. Click HERE to return to the list of problems. To put this rule into context, let’s take a look at an example: h(x) = sin(x3). If z = f(x,y) = xexy, then the partial derivatives are ∂z ∂x = exy +xyexy (Note: Product rule (and chain rule in the second term) ∂z ∂y = x2exy (Note: No product rule, but we did need the chain rule) 4. For [Math Processing Error] h ( x) = f ( g ( x)), let [Math Processing Error] u = g ( x) and [Math Processing Error] y = h ( x We need an easier way, a rule that will handle a composition like this. The chain rule is a very useful tool used to derive a composition of different functions. Suppose we need to find the first derivative of the following function: h (x)= (2x^2+3)^ {10} h(x) = (2x2 +3)10. The method of solution involves an application of the chain rule. Whether you want to find a derivative using the limit definition or using some of the many techniques of differentiation, such as the power, quotient or chain rules, Wolfram|Alpha has you covered. Then (This is an acceptable answer. 7 Example Find the derivative of f(x) = eee x. Proving the chain rule. Unit 4 Applications of derivatives. May 11, 2017 · This calculus video tutorial explains how to find derivatives using the chain rule. For example, suppose we wish to find the derivative of the function shown below. A short cut for implicit differentiation is using the partial derivative (∂/∂x). Most problems are average. However, we rarely use this formal approach when applying the chain Let's dive into the process of differentiating a composite function, specifically f(x)=sqrt(3x^2-x), using the chain rule. The chain rule starts with a composite function f(g(x)). So the question is, could we do this with any number that appeared in front of the x, be it 5 or 6 or 1 2, 0. 9 Chain Rule; 3. To see all my videos on the chain rule check out my website at http://MathMee Dec 12, 2023 · At this point we provide a list of derivative formulas that may be obtained by applying the chain rule in conjunction with the formulas for derivatives of trigonometric functions. d d x [ 8 x 2 + 2 x − 3] =? Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Here is another example: ∂/∂y [2xy Chain Rule; Let us discuss these rules one by one, with examples. Here “g” is a composite function therefore we can apply the chain rule. We can combine the above formula with the chain rule to get. Step 2: If y (x) is composite, then it can be written as f (g (x)) where g (x) is the inner function and f (x) is the outer function. Chain Rule Questions and Answers. The chain rule applied to the function sin(x) and p x5 1 gives For example, the derivative of x^2 is 2x, which means at any point on the curve, y is growing at a rate of two times x. For problems 1 – 3 do each of the following. • Solution 1 . Solution: Given, y = cos x 3. This is not a simple polynomial, so we can’t use the basic building block rules yet. We’ll solve this using three different approaches — but we encourage you to become comfortable with the third approach as quickly as possible, because that’s the one you’ll use to compute derivatives quickly as the course progresses. Example 1. Then multiply by the derivative of the stuff. 1. Many answers: Ex. Downloads expand_more. Next is cos x is the inner function and ln(x) denotes the outer function. Chain Rule - Illinois Institute of Technology Constant Factor Rule Constants come out in front of the derivative, unaffected: $$\dfrac{d}{dx}\left[c f(x) \right] = c \dfrac{d}{dx}f(x) $$ For example, $\dfrac{d Jun 6, 2018 · Chapter 3 : Derivatives. To avoid confusion, we ignore most of the subscripts here. Example: Let us compute the derivative of sin(p x5 1) for example. 2. Solutions. If w Jan 21, 2014 · The chain rule formula is as follows: (f (g (x)))’=f’ (g (x)) *g’ (x) That is, the derivative of the composition of two functions equals the derivative of the outer function times the derivative of the inner function. For instance, if f and g are functions, then the chain rule expresses the derivative of their composition. 32 now follows from the chain rule. An example of one of these types of functions is \ (f (x) = (1 + x)^2\) which is formed by taking the function \ (1+x\) and plugging it into the function \ (x^2\). e. For x y3 = 1 x y 3 = 1 do each of the following. Solution The chain rule is used twice, each time with outside function the exponential function: f0(x) = d dx h eee x i = eee x d dx h eex i = eee x eex ex: 21. Examples. Exponent and Logarithmic - Chain Rules a,b are constants. The quotient rule is a very useful formula for deriving quotients of functions. 12 Higher Order Derivatives; 3. Let’s start with an example to see how it works ( click here ): The derivative in Equation 3. Aug 23, 2017 · Continue learning the chain rule by watching this advanced derivative tutorial. Solution: As per the power 4. The second step required another use of the chain rule (with outside function the exponen-tial function). ∫2x cos (x 2) dx = ∫cos u du. Using the chain rule of derivatives, we find that its derivative is: f' (x)=8x (x^2+1)^3 f ′(x) = 8x(x2 + 1)3. The chain rule is defined as the derivative of a composition of at least two different types of functions, such as: y’ = \frac {d} {dx} [f \left ( g (x) \right)] y’ = dxd [f (g(x))] where g ( x) is a domain of the function f ( u ). 3. A series of free Engineering Mathematics video lessons. Rewrite. Nov 16, 2022 · Quotient Rule. Later on, you’ll need the chain rule to compute the derivative of p 4x2 + 9. For example, sin (x²) is a composite function due to the fact that its construction can take place as f (g (x)) for f (x)=sin (x) and g (x)=x². A few are somewhat challenging. In fact, in this particular case, we can just open the braces using binomial theorem and obtain a polynomial, derivative of which is easily found due to table Nov 10, 2020 · In Chain Rule for One Independent Variable, the left-hand side of the formula for the derivative is not a partial derivative, but in Chain Rule for Two Independent Variables it is. Then . Periodic Table. Answer. Anyway, the chain rule says if you take the derivative with respect to x of f(g(x)) you get f May 13, 2020 · Step-by-step math courses covering Pre-Algebra through Calculus 3. Example 1 Find the derivative \( f'(x) \) given \[ f(x) = 4 \cos (5x - 2) \] Solution to Example 1 Hence, through the chain rule, we have h ′ ( x) = 160 ( 8 x − 10) 3. 5 Describe the proof of the chain rule. In the following discussion and solutions the derivative of a function h ( x) will be denoted by or h ' ( x) . Summary of the chain rule. The Chain Rule can be extended to any finite number of functions by the above technique. First we'll take the derivative of the square-root function. Example Find d dx (e x3+2). Let’s look at how chain rule works in combination with trigonometric functions. Scientific Calculator. If y = cos x 3, find dy/dx. Suppose $$\displaystyle h(x) = \sin(x^2)$$. Solution: Assume x 2 = u ⇒ 2x dx = du. g(x, y) = sin(x2y − 2x + 4) Solution: a. 5 or for that matter n ? So let’s have a look at another example. Physics Constants. There's a differentiation law that allows us to calculate the derivatives of functions of functions. We can also call the function f as the external function and the function g as the internal function. Step 6: Simplify the chain rule derivative. Apply the quotient rule. Critical thinking question: 13) Give a function that requires three applications of the chain rule to differentiate. Then, the chain rule is used to obtain a derivative of y with respect to x. 1. The chain rule allows us to use substitution to differentiate any function in the form. The chain rule. The Chain Rule Using Leibniz’s Notation. 21. Let us solve a few examples to understand the calculation of the derivatives: Example 1: Determine the derivative of the composite function h (x) = (x 3 + 7) 10. Derivatives. Now apply the product rule twice. ⁡. Find the derivative of y = (4x3 + 15x)2. Let u = ax + b, then y = u½. GET STARTED. If y = f (g (x)), then as per chain rule the instantaneous rate of change of function ‘f’ relative to ‘g’ and ‘g’ relative to x results in an instantaneous rate of change of ‘f’ with respect to ‘x’. To avoid using the chain rule, recall the trigonometry identity , and first rewrite the problem as . Solution: This is a composition of three functions f(g(h(x))), where h(x) = x5 1, g(x) = p x and f(x) = sin(x). Function Derivative y = ex dy dx = ex Exponential Function Rule y = ln(x) dy dx = 1 x Logarithmic Function Rule y = a·eu dy dx = a·eu · du dx Chain-Exponent Rule y = a·ln(u) dy dx = a u · du dx Chain-Log Rule Ex3a. Put both parts into the chain rule. If we apply another function to that function, we have another multiplier applied to the first one. SOLUTION 6 : Differentiate . 3 Apply the chain rule and the product/quotient rules correctly in combination when both are necessary. An-swer. If the two functions f (x) f ( x) and g(x) g ( x) are differentiable ( i. That's the essence of why the chain rule works the way it does. Let y = cos u and u = x 3. x2 +y3 =4 x 2 + y 3 = 4 Solution. dy = f’ (x) Δx. Find y′ y ′ by solving the equation for y and differentiating directly. To avoid using the chain rule, first rewrite the problem as . As you can probably imagine, the multivariable chain rule generalizes the chain rule from single variable calculus. The exponential function f (x) = e x has the property that it is its own derivative. Here, we will learn how to find the derivatives of parametric equations. x y3 =1 x y 3 = 1 Solution. It states that the derivative of a constant function is zero; that is, since a constant function is a horizontal line, the slope, or the rate of change, of a constant function is \ (0\). Step 3: Find the derivative of the outer function, leaving the inner function. Calculate the derivative \(\dfrac{dy}{dx}\) for each of the following parametrically defined plane curves, and locate any critical points on their respective graphs. ln y = x ln b . The reason is that, in Chain Rule for One Independent Variable, \(z\) is ultimately a function of \(t\) alone, whereas in Chain Rule for Two Independent Variables 3. Example 3. Unit 1 Limits and continuity. Nov 2, 2020 · Example \(\PageIndex{1}\): Finding the Derivative of a Parametric Curve. In the figure, Δy = CB = (y + Δy) – y = f (x In this video, we will continue learning how to calculate the derivative of a function raised to a power using the =chain rule combined with power rule or wh Mar 17, 2024 · Search. To calculate ∂ f / ∂ x, treat the variable y as a constant. Using implicit differentiation, again keeping in mind that ln b ln b is constant, it follows that 1 y d y d x = ln b . Solution: Now, let u = x 3 + 7 = g (x), here h (x) can be written as h (x) = f (g Applying the product rule is the easy part. this quantity determines the approximate change in f (x) due to the change in x from x to x + Δx, as shown in the figure below. The product rule is a very useful tool for deriving a product of at least two functions. Unit 8 Applications of integrals. 14. Use the chain rule to calculate h (x), where h(x) = f(g(x)). Chain rule with trig functions. When taking the derivatives of \(y\) terms, the usual rules apply except that, because of the Chain Rule, we need to multiply each term by \(y^\prime \). Scroll down the page for examples and solutions on how to use the rules. More importantly, we will learn how to combine these differentiations for more complex functions. Even performing implicit differentiation, finding partial derivatives and finding the value of a derivative at a point are broken down and explained The following table shows the derivative or differentiation rules: Constant Rule, Power Rule, Product Rule, Quotient Rule, and Chain Rule. The Chain Rule. However, we rarely use this formal approach when applying the chain Oct 24, 2023 · The following steps are used in order to find the derivative of a composite function y (x) using chain rule: Step 1: First check that y (x) is a composite function or not. Consider a single variable function y = f (x); the total derivative of the function is given by. Step 4: Find the derivative of the inner function. The last operation that you would use to evaluate this expression is multiplication, the product of 4x2 9 and p 4x2 + 9, so begin with the product rule. Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex equations without much hassle. It is a rule that states that the derivative of a product of two functions is equal to the first function f(x) in its original form multiplied by the derivative of the second function g(x) and then added to the original form of the second function g(x) multiplied by the the product rule and the chain rule for this. Derivative of aˣ (for any positive base a) Derivative of logₐx (for any positive base a≠1) Worked example: Derivative of 7^ (x²-x) using the chain rule. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. Identify the inner and outer functions of the following composite function then find the derivative of h ( x) = 2 cos 3. du/dx = 3x 2. 11 Related Rates; 3. function. 2. Calculate ∂ f / ∂ x and ∂ f / ∂ y for the following functions by holding the opposite variable constant then differentiating: f(x, y) = x2 − 3xy + 2y2 − 4x + 5y − 12. Keep in mind that everything we’ve learned about power rule, product rule, and For example, y = cosx x2 We write this as y = u v where we identify u as cosx and v as x2. Discuss and solve an example where we calculate partial derivative. 1 y d y d x = ln b . Unit 5 Analyzing functions. Now apply the product rule. The chain rule for this case is, dz dt = ∂f ∂ We’ll illustrate in the problems below. The derivative of the outer function is equivalent to\(\frac{1}{\cos x}\). Step 5: Multiply the results from step 4 and step 5. We will use several examples and practice problems. Solution: The derivatives of f and g are f (x) = 6 g (x) = − 2. Instead, we use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. The chain rule is a method for finding the derivative of composite functions, or functions that are made by combining one or more functions. As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. Nov 16, 2022 · Section 3. 2 Apply the chain rule together with the power rule. In an internal combustion engine, as a piston moves downward the connecting rod rotates the crank in the clockwise direction, as shown in Figure [fig:crank] below. Differentiate using the chain rule, which states that d dx [f (g(x))] d d x [ f ( g ( x))] is f '(g(x))g'(x) f ′ ( g ( x)) g ′ ( x) where f (x) = x6 f ( x) = x 6 and g(x) = 4x−14 g ( x) = 4 x - 14. Learn more about the chain rule of differentiation here. This notation for the chain rule is used heavily in physics applications. Solution Again, we use our knowledge of the derivative of ex together with the chain rule. Differentiate both parts separately. Let a > 0 and set f(x) = ax — this is what is known as an exponential function. The reason is that we can chain even more functions together. Now, using Nov 20, 2021 · The first of these is the exponential function. 8 Derivatives of Hyperbolic Functions; 3. Step-by-Step Examples. Then we calculate the derivative of y(t) with respect to t. Unit 2 Derivatives: definition and basic rules. It's called the Chain Rule, although some text books call it the Function of a Function Rule. He then goes on to apply the chain rule a second time to what is inside the parentheses of the original expression. We're going to break this up into two steps. 4 Product and Quotient Rule; 3. Example: Solve ∫2x cos (x 2) dx. Use the chain rule to find Dec 29, 2020 · Take the derivative of each term in the equation. Chain rule in differentiation is defined for composite functions. (4x − 14)6 ( 4 x - 14) 6. The formula for the chain rule of integrals is as follows: We can understand this formula by considering the function f (x)= (x^2+1)^4 f (x) = (x2 + 1)4. The chain rule applied to the function sin(x) and p x5 1 gives cos(p x 5 The name ”chain rule” is because we can chain even more functions together: Problem: Let us compute the derivative of sin(√ x5 −1) for example. Substitute u = ax + b back into your answer. 2: Calculating Partial Derivatives. 10 Implicit Differentiation; 3. The chain rule tells us how to find the derivative of a composite function: d d x [ f ( g ( x))] = f ′ ( g ( x)) g ′ ( x) The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's Calculus Examples. Discuss and prove an identity involving partial derivatives. The Chain Rule is a common place for students to make mistakes. Check that the derivatives in (a) and (b) are the same. Calculus. In this case we are going to compute an ordinary derivative since z really would be a function of t only if we were to substitute in for x and y. 27 In differential calculus, the chain rule is a formula used to find the derivative of a composite function. Let us see an example and solve an integral using this antiderivative rule. By breaking down the function into its components, sqrt(x) and 3x^2-x, we demonstrate how their derivatives work together to make differentiation easier. Derivatives - Quotient and Chain Rule and Simplifying. and. Nov 16, 2022 · Case 1 : z = f(x, y), x = g(t), y = h(t) and compute dz dt. Find the derivative of y = 6e7x+22 Answer: y0 = 42e7x+22 a = 6 u Mar 24, 2023 · In Chain Rule for One Independent Variable, the left-hand side of the formula for the derivative is not a partial derivative, but in Chain Rule for Two Independent Variables it is. d dx (ex3 Geometrical Interpretation of Total Derivatives. This lesson contains plenty of practice problems including examples of c Derivatives of composite functions in one variable are determined using the simple chain rule formula. For example: Consider a function: g (x) = ln (sin x) g is a composite function. 13. 7 Derivatives of Inverse Trig Functions; 3. The single variable chain rule tells you how to take the derivative of the composition of two functions: d d t f ( g ( t)) = d f d g d g d t = f ′ ( g ( t)) g ′ ( t) May 4, 2023 · Step 6: Simplify the obtained chain rule derivative. This method can be used for any fractional power of any linear or non-linear expression. Hint. 6 Derivatives of Exponential and Logarithm Functions; 3. Get all the \(y^\prime \) terms on one side of the equal sign and put the remaining terms on the other side. Solution: This is a composition of three functions f(g(h(x))), where h(x) = x5 −1, g(x) = √ xand f(x) = sin(x). This means that the slope of a tangent line to the curve y = e x at any point is equal to the y-coordinate of the point. Find y′ y ′ by implicit differentiation. x as h ( x) = 2 ( cos. In other words, the chain rule helps in differentiating *composite functions*. The chain rule states formally that. Worked example: Derivative of log₄ (x²+x) using the chain rule. The quotient rule The rule states: Key Point Thequotientrule:if y = u v then dy dx = vdu dx −udv v2 Apr 22, 2024 · This is the same as the chain rule. In implicit differentiation this means that every time we are differentiating a term with \(y\) in it the inside function is the \(y\) and we will need to add a \(y'\) onto the term since that will be Jun 21, 2023 · The chain rule states that the derivative of this new function with respect to \(x\) is the product of derivatives of the individual functions. (Note: We used the chain rule on the first term) ∂z ∂y = 30y 2(x +y3)9 (Note: Chain rule again, and second term has no y) 3. 5 Derivatives of Trig Functions; 3. Substitute this into the integral, we have. From the chain rule dy dx = dy du × du dx = cosu× 5 = 5cos5x Notice how the 5 has appeared at the front, - and it does so because the derivative of 5x was 5. Formula for the chain of integration. This is one of the most common rules of derivatives. Unit 7 Differential equations. This case is analogous to the standard chain rule from Calculus I that we looked at above. Note that the numerator of the quotient rule is very similar to the product rule so be careful to not mix the two up! The Example 2. x. Here are a set of practice problems for the Derivatives chapter of the Calculus I notes. It will also handle compositions where it wouldn't be possible to multiply it out. 3. The engineer's function \(\text{wobble}(t) = 3\sin(t^3)\) involves a function of a function of \(t\). As we will quickly see, each derivative rule is necessary and useful for finding the instantaneous rate of change of various functions. df dx = lim h → 0 f(x + h) − f(x) h = lim h → 0 ax + h − ax h = lim h → 0ax ⋅ ah Nov 16, 2022 · 3. Download Page (PDF) Download Full Book (PDF) Resources expand_more. Differentiate . 1 State the chain rule for the composition of two functions. In this presentation, both the chain rule and implicit differentiation will Nov 17, 2020 · Example 13. It is a rule that states that the derivative of a quotient of two functions is equal to the function in the denominator g(x) multiplied by the derivative of the numerator f(x) subtracted from the numerator f(x) multiplied by the derivative of the denominator g(x), all divided by the The rule for differentiating constant functions is called the constant rule. Example am assuming familiarity with the chain rule. Example Aug 29, 2023 · Notice that the 3 derivatives are linked together as in a chain (hence the name of the rule). In this case, y is treated as a constant. It is a product, so we could write it as y = (4x3 + 15x)2 = (4x3 + 15x)(4x3 + 15x) and use the product rule. Solution: This is a composition of three functions f(g(h(x))), where h(x) = x5 1, g(x) = p xand f(x) = sin(x). 5. \(F_1(x) = (1-x)^2\): Feb 15, 2021 · Example – Combinations. The reason is that, in Chain Rule for One Independent Variable, \(z\) is ultimately a function of \(t\) alone, whereas in Chain Rule for Two Independent Variables The antiderivative chain rule is used if the integral is of the form ∫u' (x) f (u (x)) dx. dy/du = -sin u . Use the Chain Rule to find the derivatives of the following functions, as given in Example 59. If x is a variable and is raised to a power n, then the derivative of x raised to the power is represented by: d/dx(x n) = nx n-1. Example 59 ended with the recognition that each of the given functions was actually a composition of functions. Google Classroom. bl my zr nd ue qu rd wh vh re