Derivatives of Logarithmic and Exponential functions.





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

f(x)=ax , where a is a constant and “x” is a variable.

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

The logarithmic functions are the inverse function of exponential function.

Or , if ” y = f(x) = ax ” 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) = ex 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 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,

\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 =ex 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 = ax is a exponential function in base “a” .

As we know a= “e log a We can rewrite the function y = ax 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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