# Derivative of e^4x from First Principle

The derivative of e^4x with respect to x is 4e^4x. In this post, we will evaluate the derivative of e to the power 4x from first principle.

## Derivative of e^4x by first principle

Let f(x) be a function in one variable x. The derivative of f(x) from first principle is given by the following limit:

d/dx(f(x)) =lim_{h to 0} frac{f(x+h)-f(x)}{h} quad cdots (i)

Take f(x)=e^{4x} in the above formula (i). So the derivative of e^{4x} by first principle is equal to
d/dx(e^{4x}) =lim_{h to 0} frac{e^{4(x+h)}-e^{4x}}{h}
=lim_{h to 0} frac{e^{4x+4h}-e^{4x}}{h}
=lim_{h to 0} frac{e^{4x} cdot e^{4h}-e^{4x}}{h} using the rule of indices e^{a+b} =e^acdot e^b
=lim_{h to 0} frac{e^{4x}(e^{4h}-1)}{h}
=e^{4x} lim_{h to 0} frac{e^{4h}-1}{h} (As the function e^{4x} is independent of h. So one can take it out of the limit)
=e^{4x} lim_{h to 0} frac{e^{4h}-1}{4h} times 4
Let 4h=t. So t to 0 as h tends to zero. So the above limit is
=4e^{4x} lim_{t to 0} frac{e^t-1}{t}
=4e^{4x} times 1 using the limit formula lim_{x to 0} frac{e^{x}-1}{x}=1
=4e^{4x}
Thus the derivative of e to the power 4x is 4e^4x and this is obtained by the first principle of derivatives.

## Question Answer on Derivative of e^4x

Question 1: Find the derivative of e^4x at x=0 from first principle.
The derivative of e^4x from the first principle is 4e^{4x} (see the above proof).
[d/dx(e^{4x})]_{x=0} =[4e^{4x}]_{x=0} =4e^{4cdot 0} =4e^0 =4 times 1 =4
Note that e^4 is a constant number as the number e is so. We know that the derivative of a constant is zero (For a proof of this fact, click the page on Derivative of a constant is 0). So we conclude that the derivative of e to the power 4 is zero.