The Chain Rule

The chain rule allows us to take the derivative of compound functions like $\sin\left(x^2\right)$ or $\bigl(\sin(x)\bigr)^2$. If $f(x)=\sin(x)$ and $g(x)=x^2$, then $\sin\left(x^2\right)=f\left(g(x)\right)$ and $\left(\sin(x)\right)^2 = g\left(f(x)\right)$.

Version 1 of the chain rule says:

So the derivative of $\sin\left(x^2\right)$ is $\cos\left(x^2\right)\cdot 2x$, or $2x \cos\left(x^2\right)$, while the derivative of $\sin^2(x)$ is $2 \sin(x) \cdot \cos(x)$.

It's often useful to let $u=g(x)$, so our rule becomes

Combining this with our derivatives of basic functions, we get:

In particular, when taking the derivative of $\sin\left(x^2\right)$, just let $u=x^2$, so we get the derivative of $\sin(u)$ being $\cos(u)\cdot 2x = 2x\cos\left(x^2\right)$. When taking the derivative of $\sin^2(x)$, take $u=\sin(x)$, so the derivative of $u^2$ is $\displaystyle 2 u \frac{du}{dx} = 2 \sin(x)\cos(x)$.

Version 2 of the chain rule comes from taking $y=f(u)$, where $u=g(x)$, so we have $$\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}.$$
Recall that $\displaystyle\frac{dy}{du}$ is another name for $f'(u)=f'\left(g(x)\right)$, while $\displaystyle\frac{du}{dx}$ is another name for $g'(x)$.