Waves/Derivatives

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. For a more detailed discussion and exploration of the differentiation and of Calculus, see Calculus and Differentiation.

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.

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

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


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

and


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