# Derivatives of Logarithmic and Exponential functions.

Exponential functions are the function which are defined in the form of:

f(x)=a^{x} , where a is a constant and “x” is a variable.

The function “f(x) = a^{x}“^{ }is called an exponential function in base “a”.

The logarithmic functions are the inverse function of exponential function.

Or , if ” y = f(x) = a^{x} ” then , x=f^{-1}(y) is called the logarithmic function and is denoted by

y= log_{ a} x , which is called the logarithmic function in base “a”.

And , The natural exponential is the exponential function when the function is in the base of a constant “e”.

or, f(x) = e^{x} is called natural exponential function , where the constant “e” is defined by :

And the value of “e” is , an irrational number which value is approx.” 2.71828182845904523536 ”

The inverse of natural exponential function is called natural logarithmic function , which is defined by:

y = log _{e} x

for ease the natural logarithmic function is also written by excluding the base “e” ( log x) and

also by replacing log with ln ( ln x).

**Derivative of Natural Logarithmic function**:

By the definition of derivative:

Now using the properties of logarithms:

Now, If we replace by “v” or ,

Then, as ,

Now , as

we can write above equation as:

And as , natural logarithm of “e” is 1.

Thus:

The derivative natural logarithmic function is:

**Derivative of Logarithmic function**:

If “y= log _{a} x” is a logarithmic function in base “a”.

We can also re-write the function as: or , y = log _{a} e . log x, by using the properties of logarithms .

And as “a” and “e” both are constants “log _{a }e” will also be a constant so while differentiating we can take the “log _{a }e” out of the differentiation as “log _{a }e” is a constant.

So,

And as we have already derived the derivative of natural logarithms, we can differentiate the natural logarithm in the equation which give us:

**Derivative of Natural Exponential function**:

We know , y =e^{x} is the natural exponential function.

We can also write it’s inverse function as: x = log y

Now let’s differentiate both side of “x = log y” with respect to “x”:

Now using the chain rule:

We cam re-arrange above equation as:

Thus we found the derivative of natural exponential function which is:

or ,

**Derivative of Exponential function**:

If y = a^{x} is a exponential function in base “a” .

As we know a= “e ^{log a }“^{ }We can rewrite the function y = a^{x} as:

y= e^{ x log a}

So,

Thus the derivative of exponential function is found to be:

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