Suppose you are asked to differentiate the function
$$ F(x)=\sqrt{x^2+1} $$
The differentiation formulas you learned in the previous sections of this chapter do not enable you to calculate $F'(x)$. Observe that $F$ is a composite function. In fact, if we let $y=f(u)=\sqrt(u)$ and let $u=g(x)=x^2+1$, then we can write $y=F(x)=f(g(x))$.
We know how to differentiate both $f$ and $g$, so it would be useful to have a rule that tells us how to find the derivative of $F=f\circ g$ in terms of the derivatives of $f$ and $g$.
It turns out that the derivative of the composite function $f\circ g$ is the product of the derivatives of $f$ and $g$. This fact is one of the most important of the differentiation rules and is called the Chain Rule.
It seems plausible if we interpret derivatives as rates of change. Regard $du/dx$ as the rate of change of $u$ with respect to $x$, $dy/du$ as the rate of change of $y$ with respect to $u$, and $dy/dx$ as the rate of change of $y$ with respect to $x$. If $u$ changes twice as fast as $x$ and $y$ changes three times as fast as $u$, then it seems reasonable that $y$ changes six times as fast as $x$, and so we expect that $dy/dx$ is the product of $dy/du$ and $du/dx$.
If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then the composite function $F=f\circ g$ defined by $F=f(g(x))$ is differentiable at $x$ and $F'$ is given by the product
$$ F'(x)=f'(g(x))\cdot g'(x) $$
<aside>
Find $F'(x)$
$$ F(x)=\sqrt{x^2+1} $$
</aside>
<aside>
Differentiate
$$ y=\sin(x^2) $$
$$ y=\sin^2(x) $$
</aside>
If $n$ is any real number and $u = g(x)$ is differentiable, then
$$ \frac{d}{dx}(u^n)=nu^{n-1}\frac{du}{dx} $$
$$ \frac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}\cdot g'(x) $$
<aside>
Differentiate
$$ y=(x^3-1)^{100} $$
</aside>
<aside>
Differentiate
$$ y=\frac{1}{\sqrt[3]{x^2+x+1}} $$
</aside>
<aside>
Differentiate
$$ g(t)=\left(\frac{t-2}{2t+1}\right)^9 $$
</aside>