A hybrid chain rule

In addition the previous iterations of the chain rule, there is also a useful hybrid form, given below, where we assume $u$ is a function of $x$.

Applications

This form allows us to expand the scope of many of our derivative formulas.  In the following, we assume $u$ is a function of $x$.

• The derivative of $x^n$ (with respect to $x$) is $n x^{n-1}$, so $\frac{d}{dx}(u^n)=n u^{n-1} \frac{du}{dx}$. (Notice we are differentiating with respect to $x$, but our variable is $u$, thus we need the chain rule.)
  

           DO:  Consider $\frac{d}{dx}((3x-2)^{5})$  What is u?  Find this derivative.

• The derivative of $\sin(x)$ is $\cos(x)$, so $\frac{d}{dx}\sin u=\cos(u) \frac{du}{dx}$.

• The derivative of $\cos(x)$ is $-\sin(x)$.           DO:  What is $\frac{d}{dx}\cos(u)$?

• The derivative of $e^x$ is $e^x$, so the derivative of $e^u$ is $e^u \frac{du}{dx}$.