The derivative of e^{4x} is 4e^{4x}. Note that e^{4x} is an exponential function with exponential 4x. Here, we will find the derivative of e to the power 4x using the following rules:

- Logarithmic differentiation
- First principle of derivatives
- Chain rule of derivatives.

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## Derivative of e^{4x} Formula

The derivative of e^{4x }is 4e^{4x}. Mathematically, we can write it as

d/dx(e^{4x}) = 4e^{4x } or (e^{4x})’ = 4e^{4x}.

## What is the derivative of e^{4x}?

**Answer:** The derivative of e^{4x }is 4e^{4x}.

*Proof:* By the logarithmic differentiation, we will find the derivative of e^{4x}. Let us assume that

y = e^{4x}

Taking logarithms with base e to both sides, we obtain that

log_{e} y = log_{e} e^{4x}

⇒ log_{e} y = 4x by the logarithm rule log_{e} e^{a} = a.

Differentiating with respect to x, we get that

$\dfrac{1}{y} \dfrac{dy}{dx}=4$

⇒ $\dfrac{dy}{dx}=4y$

⇒ $\dfrac{dy}{dx}=4e^{4x}$

This shows that the derivative of e^{4x} is 4e^{4x} and this is obtained by the logarithmic differentiation.

**Also Read:**

Derivative of e: The derivative of e^{sin x}^{sin x} is cos x e^{sin x}.Integration of modulus of x: The integration of mod x is -x|x|/2+c.Derivative of 1/x: The derivative of 1 by x is -1/x^{2}. |

## Derivative of e^{4x} by Chain Rule

To find the derivative of a composite function, we use the chain rule. We will now find the derivative of e to the power 4x by the chain rule.

Let z=4x.

d/dx(e^{4x}) | = d/dz(e^{z}) × d/dx(4x) |

= e^{z} × 4 | |

= 4e^{z} | |

= 4e^{z} as z=4x. |

## Derivative of e^{4x} by First Principle

By the first principles, the derivative of a function f(x) is given by the following limit:

$\dfrac{d}{dx}(f(x))=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

Put f(x)=e^{4x}.

So the derivative of e^{4x} by first principle is

$\dfrac{d}{dx}(e^{4x})= \lim\limits_{h \to 0} \dfrac{e^{4(x+h)}-e^{4x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{4x+4h}-e^{4x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{4x} \cdot e^{4h}-e^{4x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{4x}(e^{4h}-1)}{h}$

=e^{4x} $\lim\limits_{h \to 0} \Big(\dfrac{e^{4h}-1}{4h} \times 4 \Big)$

= 4e^{4x} $\lim\limits_{h \to 0} \dfrac{e^{4h}-1}{4h}$

[Let t=4h. Then t→0 as x →0]

= 4e^{4x} $\lim\limits_{t \to 0} \dfrac{e^{t}-1}{t}$

= 4e^{4x} ⋅ 1

= 4e^{4x}

∴ The differentiation of e^{4x} is 4e^{4x} and this is achieved from the first principle of derivatives.

## FAQs on Derivative of e^{4x}

**Q1: What is the derivative of e**

^{4x}?Answer: The derivative of e^{4x} is 4e^{4x}.

**Q2: What is the integration of e**

^{4x}?Answer: The integration of e^{4x} is e^{4x}/4+c.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.