I am having some issues with Derivatives. My first question is what is the derivative of e^4x? I know one of the rules is (e^x)' = e^x. Then why is (e^4x)' not e^4x?
Thanks
Adam
Another (easier) way to write the chain rule is $\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \end{align*}$. So you let your "inner" function of x be u, so that you can then write your function y in terms of u, take the derivative of each, then convert $\displaystyle \displaystyle \begin{align*} \frac{dy}{du} \end{align*}$ back to a multiple of x and multiply them.
For example, if our function was $\displaystyle \displaystyle \begin{align*} y = (x + 3)^2 \end{align*}$, our "inner" function is $\displaystyle \displaystyle \begin{align*} u = x + 3 \end{align*}$ leaving $\displaystyle \displaystyle \begin{align*} y = u^2 \end{align*}$.
Differentiating each gives $\displaystyle \displaystyle \begin{align*} \frac{du}{dx} = 1 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} \frac{dy}{du} = 2u = 2(x + 3) \end{align*}$.
Therefore the derivative is $\displaystyle \displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2(x + 3) \cdot 1 = 2(x + 3) \end{align*}$.