Euler's number

Before getting into the derivatives of other special functions, we must first learn about Euler's number.

Euler's number is a mathematical constant that has special properties when regarding the derivative of functions involving it.

Take a look at the following graphs of $y = 2^x$ and $y = 4^x$:

observation

As you can see, the derivative of $y = 2^x$ is below the original function, while the derivative of $y = 4^x$ is above the original function.

This means that there must be a value for the base of the exponent such that its derivative is the exact same function as itself!

This base has the symbol $e$ and it is known as Euler's Number and it is special because its derivative is the exact same function.

The approximate value for Euler's Number is $e \approx 2.718281...$

The natural exponential function

The function has the following form:

$f(x) = e^x$

Take a look at the following graph of the function $y = e^x$ and its derivative:

This illustrates that the graph of $f(x) = e^x$ is the same as its derivative $f'(x)$. In fact,

$f(x) = e^x = f'(x) = f''(x) = f'''(x) = ... = f^{(\infty)}(x)$

No matter how many times you differentiate the natural exponential function, it will always be the same!

The natural logarithm function

The natural logarithm is the inverse of the natural exponential function. It's base is Euler's Number, however, since it is such a common function used in math, it has a special notation shown below along with its derivative:

$f(x) = log_{e} x = \ln x$

$\frac {d} {dx} [ \ln x ] = \frac {1} {x}$

Special functions and their derivatives

We will now explore the derivatives of the following functions:

• Exponential functions of the form $y = a^x$

• Logarithmic functions of the form $y = log_{n} x$

• Trigonometric functions: $y = \sin x$, $y = \cos x$ and $y = \tan x$

Derivative of exponential functions

The derivative of any exponential function follows this formula:

$\frac {d} {dx} [ a^x ] = a^x \ln a$

For example, the derivative of $f(x) = 5^x$ is $f'(x) = 5^x \ln 5$

Derivative of logarithmic functions

The derivative of any logarithmic function follows this formula:

$\frac {d} {dx} [ \log_{a} x ] = \frac {1} {x \ln a }$

For example, the derivative of $f(x) = \log_{10} x$ is $f'(x) = \frac {1} {x \ln 10}$

Derivatives of sinosoidal functions

The derivatives of $\sin x$ and $\cos x$ are as follows:

$\frac {d} {dx} [ \sin x ] = \cos x$

$\frac {d} {dx} [ \cos x ] = -\sin x$

So, for $f(x) = \sin x$:

$f'(x) = \cos x$

$f''(x) = -\sin x$

$f'''(x) = -\cos x$

$f^{(4)}(x) = -(-\sin x) = \sin x$

Note that $f(x) = f^{(4)}(x)$. This applies to both $\sin x$ and $\cos x$.

Derivative of the tangent function

We can use the fact that $\tan x = \frac {\sin x} {\cos x}$, as well as the derivatives of $\sin x$ and $\cos x$ to find the derivative of $\tan x$.

$\frac {d} {dx} [ \tan x ] = \frac {d} {dx} [ \frac {\sin x} {\cos x} ]$

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

$\quad \quad \quad \quad = \frac {(\cos x)(\cos x) \ - \ (\sin x)(-\sin x)} {\cos^2 x}$

$\quad \quad \quad \quad = \frac {\cos^2 x \ + \ \sin^2 x} {\cos^2 x}$

$\quad \quad \quad \quad = \frac {1} {\cos^2 x}$

$\quad \quad \quad \quad = \sec^2 x$

So, the derivative of $\tan x$ is:

$\frac {d} {dx} [\tan x] = \sec^2 x$