# Differentiation of Exponential and Logarithmic Functions

The exponential function
derivative is not easily solved directly with the definition of the derived
function. However, it can be solved using the inverse function derivative,
namely the logarithmic function. So, in this article the derivative of the
logarithmic function will take precedence before obtaining the derivative
function derivative.

### Derivatives of Logarithmic Functions

For example, a function f is stated by the following formula,

f(x) =

^{a}log x, x > 0
then the derivative of the function can be obtained by the following steps,.

Note that for h → 0, then u → 0.

So, the function f(x) =

^{a}log x, x > 0 is differentiable and the derivative is expressed by a formula.
Especially for the logarithmic function with a base number e expressed by f(x) = ln x, x > 0, the derivative formula is as follows.

In general, the derivative
of function f(x) =

^{a}log h(x) is
The function derivative f (x) = ln g (x) is

###
Derivative Function f(x) =
e^{x}

To obtain a function
formula in the form of f(x) = e

^{x}, we can use the derivative formula of the logarithmic function. However, we can also use the inverse function f(x) = ln x first. Suppose y = e^{x}then applies x = ln y.
By using

**Leibniz**notation in derivatives, the following form is obtained,
So, the derivative of the
exponential function f(x) = e

^{x}is expressed by the formula:
f’(x) = e

^{x}^{}

Whereas, for exponential functions with numbers and past bases, namely f(x) = a

^{x}in the same way as above we can find the formula as follows.
f’(x) = a

^{x n}log a or f’(x) = a^{x}ln a
In general, the derivative
of the function f(x) = e

^{h(x)}is
f’(x)’=h’(x)

**.**e^{h(x)}
the derivative of function
f(x) = a

^{g(x)}is
f’(x) = a

^{g(x)}**.**g’(x) ln a**Example 1.**

Find the function
derivative f(x) =

^{3}log (2x – 4).
Answer:

**Example 2.**

Find the function derivative f(x) = ln (2x

^{3}+ 7).
Answer:

**Example 3.**

Find the function derivative f(x) = log 3x.

Answer:

**Example 4.**

Find the function derivative f(x) = e

^{2x +3}
Answer:

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