Differentiation of Exponential Function

Here you will learn differentiation of exponential function by using first principle and its examples.

Let’s begin –

Differentiation of Exponential Function

(1) Differentiation of $e^x$ :

The differentiation of $e^x$ with respect to x is $e^x$.

i.e. $d\over dx$ $e^x$ = $e^x$

Proof Using first Principle :

Let f(x) = $e^x$. Then, f(x + h) = $e^{x + h}$

$\therefore$   $d\over dx$(f(x)) = $lim_{h\to 0}$ $f(x + h) – f(x)\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $e^{x + h} – e^x\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $e^x.e^h – e^x\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $e^x$ ($e^h – 1\over h$)

$\implies$ $d\over dx$(f(x)) = $e^x$  $lim_{h\to 0}$ ($e^h – 1\over h$)

because, [$lim_{h\to 0}$($e^h – 1\over h$) = 1]

$\implies$ $d\over dx$(f(x)) = $e^x$ $\times$ 1 = $e^x$

Hence, $d\over dx$ ($e^x$) = $e^x$

Example : What is the differentiation of $e^{2x}$ ?

Solution : Let y  = $e^{2x}$

$d\over dx$ (y) = $d\over dx$ $e^{2x}$

By using chain rule,

$d\over dx$ (y) = 2$e^{2x}$

Hence, $d\over dx$ ($e^{2x}$) = 2$e^{2x}$

(2) Differentiation of $a^x$ :

The differentiation of $a^x$ with respect to x is $a^x log_e a$.

i.e. $d\over dx$ $a^x$ = $a^x log_e a$

Proof Using first Principle :

Let f(x) = $a^x$. Then, f(x + h) = $a^{x + h}$

$\therefore$   $d\over dx$(f(x)) = $lim_{h\to 0}$ $f(x + h) – f(x)\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $a^{x + h} – a^x\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $a^x.a^h – a^x\over h$

$d\over dx$(f(x)) = $lim_{h\to 0}$ $a^x$ ($a^h – 1\over h$)

$\implies$ $d\over dx$(f(x)) = $a^x$  $lim_{h\to 0}$ ($a^h – 1\over h$)

because, [$lim_{h\to 0}$($a^h – 1\over h$) = $log_e a$]

$\implies$ $d\over dx$(f(x)) = $a^x$ $\times$ $log_e a$ = $a^x$ $log_e a$

Hence, $d\over dx$ ($a^x$) = $a^x$ $log_e a$

Example : What is the differentiation of $5^{x}$ ?

Solution : Let y  = $5^{x}$

$d\over dx$ (y) = $d\over dx$ $5^{x}$

$d\over dx$ (y) = $5^x log_e 5$

Hence, $d\over dx$ ($5^{x}$) = $5^x log_e 5$

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Related Questions

What is the differentiation of $e^{sinx}$ ?

What is the integration of $e^x$ ?