1.8: Math Review — Derivatives

This section provides a quick review of the idea of the derivative. Often we are interested in the slope of a line tangent to a function y(x) at some value of x. This slope is called the derivative and is denoted dy∕dx. Since a tangent line to the function can be defined at any point x, the derivative itself is a function of x:

\[g(x)=\frac{d y(x)}{d x}\label{1.25}\]

As Figure \(\PageIndex{15}\): illustrates, the slope of the tangent line at some point on the function may be approximated by the slope of a line connecting two points, A and B, set a finite distance apart on the curve:

\[\frac{d y}{d x} \approx \frac{\Delta y}{\Delta x}\label{1.26}\]

As B is moved closer to A, the approximation becomes better. In the limit when B moves infinitely close to A, it is exact.

Derivatives of some common functions are now given. In each case a is a constant.

\[\frac{d x^{a}}{d x}=a x^{a-1}\label{1.27}\]

\[\frac{d}{d x} \exp (a x)=a \exp (a x)\label{1.28}\]

\[\frac{d}{d x} \log (a x)=\frac{1}{x}\label{1.29}\]

\[\frac{d}{d x} \sin (a x)=a \cos (a x)\label{1.30}\]

\[\frac{d}{d x} \cos (a x)=-a \sin (a x)\label{1.31}\]

\[\frac{d a f(x)}{d x}=a \frac{d f(x)}{d x}\label{1.32}\]

\[\frac{d}{d x}[f(x)+g(x)]=\frac{d f(x)}{d x}+\frac{d g(x)}{d x}\label{1.33}\]

\[\frac{d}{d x} f(x) g(x)=\frac{d f(x)}{d x} g(x)+f(x) \frac{d g(x)}{d x} \quad \text { (product rule) }\label{1.34}\]

\[\frac{d}{d x} f(y)=\frac{d f}{d y} \frac{d y}{d x} \quad(\text { chain rule })\label{1.35}\]

The product and chain rules are used to compute the derivatives of complex functions. For instance,

\(\frac{d}{d x}(\sin (x) \cos (x))=\frac{d \sin (x)}{d x} \cos (x)+\sin (x) \frac{d \cos (x)}{d x}=\cos ^{2}(x)-\sin ^{2}(x)\)

and

\(\frac{d}{d x} \log (\sin (x))=\frac{1}{\sin (x)} \frac{d \sin (x)}{d x}=\frac{\cos (x)}{\sin (x)}\).

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