Math Tutorial -- Derivatives

Figure 1.15: Estimation of the derivative, which is the slope of the tangent line. When point B approaches point A, the slope of the line AB approaches the slope of the tangent to the curve at point A.

This section provides a quick introduction to 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 $x$:

$$g(x) = \frac{d y(x)}{dx} .$$ (2.25)

As figure 1.15 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:

$$\frac{dy}{dx} \approx \frac{\Delta y}{\Delta x} .$$ (2.26)

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.

$$\frac{d x^a}{dx} = a x^{a - 1}$$ (2.27)

$$\frac{d}{dx} \exp (ax) = a \exp (ax)$$ (2.28)

$$\frac{d}{dx} \log (ax) = \frac{1}{x}$$ (2.29)

$$\frac{d}{dx} \sin (ax) = a \cos (ax)$$ (2.30)

$$\frac{d}{dx} \cos (ax) = -a \sin (ax)$$ (2.31)

$$\frac{d a f(x)}{dx} = a \frac{df(x)}{dx}$$ (2.32)

$$\frac{d}{dx} [f(x) + g(x)] = \frac{df(x)}{dx} + \frac{dg(x)}{dx}$$ (2.33)

$$\frac{d}{dx} f(x) g(x) = \frac{df(x)}{dx} g(x) + f(x) \frac{dg(x)}{dx} ~~~ \mbox{(product rule)}$$ (2.34)

$$\frac{d}{dx} f(y) = \frac{df}{dy} \frac{dy}{dx} ~~~ \mbox{(chain rule)}$$ (2.35)

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

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

and

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