Chapter 7 Differential Calculus

Section 7.2 Standard Derivatives

7.2.3 Derivatives of Special Functions

Derivatives of Trigonometric Functions

The sine function f:, xf(x)=sin(x) is periodic with period 2π. Thus, it is sufficient to consider the function on an interval of length 2π. A section of the graph for -πxπ is shown in the figure below:

As we see from the figure above, the slope of the sine function at x0 =± π 2 is f'(± π 2 )=0. The tangent line to the graph of the sine function at x0 =0 has the slope f'(0)=1. At x0 =±π, the tangent line has the same slope as the tangent line at x0 =0, but the sign is opposite. Hence, the slope at x0 =±π is f'(±π)=-1. Thus, the derivative of the sine function is a function that exhibits exactly these properties. A detailed investigation of the regions between these specially chosen points shows that the derivative of the sine function is the cosine function:
Derivatives of Trigonometric Functions 7.2.5
For the sine function f:, xf(x):=sin(x), we have


For the cosine function g:, xg(x):=cos(x), we have


For the tangent function h:{ π 2 +kπ:k}, xh(x):=tan(x), we have

h':{ π 2 +kπ:k}, xh'(x)=1+(tan(x ))2 = 1 cos2 (x) .

This last result comes from the calculation rules explained (explained below) and the definition of the tangent function as the quotient of the sine function and the cosine function.

Derivative of the Exponential Function

Info 7.2.6
The exponential function f:, xf(x):=ex =exp(x) has the special property that its derivative f' is also the exponential function, i.e. f'(x)=ex =exp(x).

Derivative of the Logarithmic Function

The derivative of the logarithmic function is given here without proof. For f: ]0;[ with xf(x)=ln(x) one obtains f': ]0;[ , xf'(x)= 1 x .