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 :
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.
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.
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 =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”:
Now using the chain rule:
We cam re-arrange above equation as:
Thus we found the derivative of natural exponential function which is:
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
Thus the derivative of exponential function is found to be:
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