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) = ^alog x, x > 0

then the derivative of the function can be obtained by the following steps,.

Note that for h → 0, then u → 0.

if it is known that   then the form above can be simplified to become,

So, the function f(x) = ^alog 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) = ^alog 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) = ³log (2x – 4).

Answer:

Example 2.

Find the function derivative f(x) = ln (2x³ + 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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