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 :

$e = \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$

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:

$\frac{d}{dx} (log x) = \displaystyle\lim_{\Delta x\to o} \dfrac{log(x+\Delta x) -log x}{\Delta x}$

Now using the properties of logarithms:

$= \displaystyle\lim_{\Delta x\to o} \frac{1}{\Delta x} \times log \frac{x+\Delta x}{x}$

$= \displaystyle\lim_{\Delta x\to o} log \left(\frac{x+\Delta x}{x}\right)^{\frac{1}{\Delta x}}$

Now, If we replace $\frac{1}{\Delta x}$ by “v” or , $v= \frac{1}{\Delta x}$

Then, as $\Delta x \to 0$ , $z \to \infty$

$\frac{d}{dx} log x = \displaystyle\lim_{z\to\infty} log \left(1+\frac{1}{z}\right)^{\frac{z}{x}}$

$= \frac{1}{x} \displaystyle\lim_{z\to\infty} log \left( 1+\frac{1}{z}\right)^z$

Now , as $e = \displaystyle\lim_{z\to\infty} \left( 1+\frac{1}{z}\right)^z$

we can write above equation as:

$= \frac{1}{x} log e$

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

$= \frac{1}{x}$

Thus:

The derivative natural logarithmic function is:

$\frac{d}{dx} log x = \frac{1}{x}$

Derivative of Logarithmic function:

If “y= log _a x” is a logarithmic function in base “a”.

We can also re-write the function as: $y = log_a e \times log x$ or , y = log _a e . log x, by using the properties of logarithms .

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

So,

$\frac{d}{dx} log_a x = log_a e \times \frac{d}{dx} ( log x)$

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

$\frac{d}{dx} log_a x = \frac{log_a e}{x}$

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”:

$\frac{d}{dx} x = \frac{d}{dx} log y$

Now using the chain rule:

$or, 1 = \frac{1}{y} \times \frac{d}{dx} y$

We cam re-arrange above equation as:

$\frac{d}{dx} y = y$

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

$\frac{d}{dx} y = y$ or , $\frac{d}{dx} e^x = e^x$

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, $\frac{d}{dx} a^x = \frac{d}{dx} e^{x log a}$

$= \frac{d}{d(x log a)} e^{x log a} \times \frac{d}{dx } x log a$

$= e^{x log a} \times log a$

$= a^x \times log a$

Thus the derivative of exponential function is found to be:

$\frac{d}{dx} a^x = a^x \times log a$

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