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\section*{2.5 Chain Rule}

\subsection*{Definition of the Chain Rule}

If \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g(x)\), then the composite function \(F=f \circ g\) defined by \(F(x)=f(g(x))\) is differentiable at \(x\) and \(F'\) is given by the product 
$$F'(x)=f'(g(x)) \times g'(x)$$
In Leibniz notation, if \(y=f(u)\) and \(u=g(x)\) are both differentiable functions, then
$$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$

\subsection*{Power Rule with the Chain Rule}

If \(n\) is any real number and \(u=g(x)\) is differentiable, the 
$$\frac{d}{dx}(u^n)=nu^{n-1}\frac{du}{dx} \Leftrightarrow \frac{d}{dx}{[g(x)]}^n=n{[g(x)]}^{n-1} \times g'(x)$$
