The chain rule states that the [[derivative]] of the function of a function $f(x) = h(g(x))$ is the derivative of the outer function times the derivative of the inner function. $f'(x) = h'(g(x)) * g'(x)$ We can use the chain rule to simplify complex functions by substituting a function within a function. For example, consider the function $f(x) = (x^2 + 2x)^5$ . This can be re-written as $h(g(x))$ where $h(g) = g^5$ and $g(x) = x^2 + 2x$. By the [[power rule]], the derivative $h'(g) = 5(g)^4$ and the derivative $g'(x) = 2x + 2$. Substituting $g(x)$ back in for $g$ in $h(g)$ we have $h'(g(x)) = 5(x^2+2x)^4$ And finally applying the chain rule formula we have $f'(x) = h'(g(x))*g'(x) = 5(x^2 + 2x)^4(2x+2)$ Let's try a more involved example. What is the derivative of $f(x) = e^{2x^2}$? We can rewrite this as $h(g) = e^g$ and $g(x) = 2x^2$. From the [[derivative of an exponential]], we know that the derivative of $e^g$ is also $e^g$, thus $h'(g(x)) = e^{2x^2}$. Thus we get $f'(x) = h'(g(x)) * g'(x) = e^{2x^2} * 4x$