![]() Which represents the slope of the tangent line at the point (−1,−32). A technique that is sometimes suggested for differentiating composite functions is to work from the “outside to the inside” functions to establish a sequence for each of the derivatives that must be taken.Įxample 1: Find f′( x) if f( x) = (3x 2 + 5x − 2) 8.Įxample 2: Find f′( x) if f( x) = tan (sec x).Įxample 5: Find the slope of the tangent line to a curve y = ( x 2 − 3) 5 at the point (−1, −32).īecause the slope of the tangent line to a curve is the derivative, you find that Here, three functions- m, n, and p-make up the composition function r hence, you have to consider the derivatives m′, n′, and p′ in differentiating r( x). If a composite function r( x) is defined as Note that because two functions, g and h, make up the composite function f, you have to consider the derivatives g′ and h′ in differentiating f( x). For example, if a composite function f( x) is defined as The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. Volumes of Solids with Known Cross Sections.Second Derivative Test for Local Extrema.The exponential rule states that this derivative is e to the power of the function times the derivative of the function. It is useful when finding the derivative of e raised to the power of a function. First Derivative Test for Local Extrema Explanation Transcript The exponential rule is a special case of the chain rule.Differentiation of Exponential and Logarithmic Functions.The derivative of f(g(x)) is the derivative of the outside function. Differentiation of Inverse Trigonometric Functions In order to take the derivative of this kind of equation, we need to use the chain rule. ![]() Limits Involving Trigonometric Functions.Take a Tour and find out how a membership can take the struggle out of learning math. Still wondering if CalcWorkshop is right for you? Get access to all the courses and over 450 HD videos with your subscription Once you have a grasp of the basic idea behind the chain rule. Let’s get to it! Video Tutorial w/ Full Lesson & Detailed Examples (Video) The chain rule is a formula to calculate the derivative of a composition of functions. So, throughout this lesson, we will work through numerous examples of the chain rule, combining our previous differentiation rules such as the power rule, product rule, and quotient rule, so that you will become a chain-rule master! In fact, we will come to see that the chain rule’s helpfulness extends beyond polynomial functions but is pivotal in how we differentiate: Thanks to the chain rule, we can quickly and easily find the derivative of composite functions - and it’s actually considered one of the most useful differentiation rules in all of calculus. Good grief! That would have been painful. Without it, we would have had to multiply the polynomial you see in blue by itself 10 times, simplify, and then use the power rule to find the derivative! The chain rule is used to find the derivatives of composite functions like (x2 + 1)3, (sin 2x), (ln 5x), e2x, and so on. Next, we multiplied by the derivative of the inside function, and lastly, we simplified. See, all we did was first take the derivative of the outside function (parentheses), keeping the inside as is.
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