#### Chapter 7 Differential Calculus

Section 7.2 Standard Derivatives

# 7.2.3 Derivatives of Special Functions

#### Derivatives of Trigonometric Functions

The sine function $f:ℝ\to ℝ$, $x\to f\left(x\right)=\mathrm{sin}\left(x\right)$ is periodic with period $2\pi$. Thus, it is sufficient to consider the function on an interval of length $2\pi$. A section of the graph for $-\pi \le x\le \pi$ is shown in the figure below:

As we see from the figure above, the slope of the sine function at ${x}_{0}=±\frac{\pi }{2}$ is $f\text{'}\left(±\frac{\pi }{2}\right)=0$. The tangent line to the graph of the sine function at ${x}_{0}=0$ has the slope $f\text{'}\left(0\right)=1$. At ${x}_{0}=±\pi$, the tangent line has the same slope as the tangent line at ${x}_{0}=0$, but the sign is opposite. Hence, the slope at ${x}_{0}=±\pi$ is $f\text{'}\left(±\pi \right)=-1$. 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 $f:ℝ\to ℝ$, $x\to f\left(x\right):=\mathrm{sin}\left(x\right)$, we have

$f\text{'}:ℝ\to ℝ , x\to f\text{'}\left(x\right)=\mathrm{cos}\left(x\right) .$

For the cosine function $g:ℝ\to ℝ$, $x\to g\left(x\right):=\mathrm{cos}\left(x\right)$, we have

$g\text{'}:ℝ\to ℝ , x\to g\text{'}\left(x\right)=-\mathrm{sin}\left(x\right) .$

For the tangent function $h:ℝ\setminus \left\{\frac{\pi }{2}+k\pi : k\in ℤ\right\}\to ℝ$, $x\to h\left(x\right):=\mathrm{tan}\left(x\right)$, we have

$h\text{'}:ℝ\setminus \left\{\frac{\pi }{2}+k\pi : k\in ℤ\right\}\to ℝ , x\to h\text{'}\left(x\right)=1+\left(\mathrm{tan}\left(x{\right)\right)}^{2}=\frac{1}{{\mathrm{cos}}^{2}\left(x\right)} .$

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 $f:ℝ\to ℝ$, $x\to f\left(x\right):=e{}^{x}=\mathrm{exp}\left(x\right)$ has the special property that its derivative $f\text{'}$ is also the exponential function, i.e. $f\text{'}\left(x\right)=e{}^{x}=\mathrm{exp}\left(x\right)$.

#### Derivative of the Logarithmic Function

The derivative of the logarithmic function is given here without proof. For $f:\text{}\right]0;\infty \left[\text{}\to ℝ$ with $x\to f\left(x\right)=\mathrm{ln}\left(x\right)$ one obtains $f\text{'}:\text{}\right]0;\infty \left[\text{}\to ℝ$, $x\to f\text{'}\left(x\right)=\frac{1}{x}$.