Derivative of e^4x: Formula, Proof by First Principle, Chain Rule

The derivative of e4x is 4e4x. Note that e4x 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.

Derivative of e4x Formula

The derivative of e4x is 4e4x. Mathematically, we can write it as

d/dx(e4x) = 4e4x  or (e4x)’ = 4e4x.

What is the derivative of e4x?

Answer: The derivative of e4x is 4e4x.

Proof: By the logarithmic differentiation, we will find the derivative of e4x. Let us assume that

y = e4x

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

loge y = loge e4x

⇒ loge y = 4x by the logarithm rule loge ea = 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 e4x is 4e4x and this is obtained by the logarithmic differentiation.

Derivative of e4x 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.

Derivative of e4x 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)=e4x.

So the derivative of e4x by first principle is

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

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

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

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

=e4x $\lim\limits_{h \to 0} \Big(\dfrac{e^{4h}-1}{4h} \times 4 \Big)$

= 4e4x $\lim\limits_{h \to 0} \dfrac{e^{4h}-1}{4h}$

[Let t=4h. Then t→0 as x →0]

= 4e4x $\lim\limits_{t \to 0} \dfrac{e^{t}-1}{t}$

= 4e4x ⋅ 1

= 4e4x

∴ The differentiation of e4x is 4e4x and this is achieved from the first principle of derivatives.

FAQs on Derivative of e4x

Q1: What is the derivative of e4x?

Answer: The derivative of e4x is 4e4x.

Q2: What is the integration of e4x?

Answer: The integration of e4x is e4x/4+c.

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