![]() An informal proof is provided at the end of the section. Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. As we determined above, this is the case for h(x)= \sin (x^3). In this article, were going to find out how to calculate derivatives for functions of functions. To find a rate of change, we need to calculate a derivative. We can take a more formal look at the derivative of h(x)= \sin (x^3) by setting up the limit that would give us the derivative at a specific value a in the domain of h(x)= \sin (x^3). The Chain Rule for Derivatives Introduction. In addition, the change in x^3 forcing a change in \sin (x^3) suggests that the derivative of \sin (u) with respect to u, where u=x^3, is also part of the final derivative. First of all, a change in x forcing a change in x^3 suggests that somehow the derivative of x^3 is involved. This chain reaction gives us hints as to what is involved in computing the derivative of \sin (x^3). We can think of this event as a chain reaction: As x changes, x^3 changes, which leads to a change in \sin (x^3). Consequently, we want to know how \sin (x^3) changes as x changes. ![]() We can think of the derivative of this function with respect to x as the rate of change of \sin (x^3) relative to the change in x. To put this rule into context, let’s take a look at an example: h(x)= \sin (x^3). 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. However, using all of those techniques to break down a function into simpler parts that we are able to differentiate can get cumbersome. ![]() When we have a function that is a composition of two or more functions, we could use all of the techniques we have already learned to differentiate it. The chain rule is a rule of expressing derivative of a function which is a combination of two functions.
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