Derivative of e^sin(pi*x) using the chain rule two times. Chain rule twice derivative.

Derivative of e^sin(pi*x) using the chain rule two times. Chain rule twice derivative.

We differentiate e^sin(pi*x) by using the chain rule twice. We start by differentiating with respect to sin(pi*x), which gives e^sin(pi*x) because the natural exponential is equal to its own derivative. Next, we tack on the derivative OF sin(pi*x). But this derivative requires a second application of the chain rule because sin(pi*x) is a function of a function. We start by differentiating with respect to pi*x, then we have to tack on the derivative OF pi*x to get the final chain rule derivative answer.