Derivative of e^2x

Derivative of e^2x is 2e^2x. Derivative refers to the process of finding the change in the exponential function with respect to the independent variable. Exponential functions are the functions that are power functions with the base of Euler’s Number.

In this article, we will learn about the derivative of e^2x and its formula including the proof of the formula using the first principle of derivatives, product rule, and chain rule in detail.

What is Derivative in Math?

Derivative of a function is the rate of change of the function to any independent variable. The derivative of a function f(x) is denoted as f′(x) or (d/dx)​[f(x)]. The differentiation of an exponential function is called a derivative of the exponential function or exponential derivatives.

What is Derivative of e^2x?

Among exponential derivatives, the derivative of e^2x is one of the derivatives. Derivative of e^2x is 2e^2x. The derivative of e^2x is the rate of change to the variable i.e., x. The resultant of the derivative of e^2x is 2e^2x. Mathematically derivative of e^2x is represented as, d/dx(e^2x) = 2e^2x

Derivative of e^2x Formula

Formula for derivative of e^2x is given by:

(d/dx) [e^2x] = 2e^2x

(e^2x)’ = 2e^2x

Proof of Derivative of e^2x

Derivative of e^2x can be proved using the following ways:

• By using First Principle of Derivative
• By using Product Rule
• By using Chain Rule

Derivative of e^2x using First Principle of Derivatives

To prove derivative of e^2x using First Principle of Derivatives, we will use basic limits and exponential formulas which are listed below:

• e^{(x + y)} = e^x.e^y
• lim_x->0 (e^x – 1)/x = 1

Let’s start proof for derivative of e^2x:

By First Principle of Derivatives,

(d/dx) e^2x = lim_x->0 [e^{(2*(x + h))} – e^(2x)]/[(x + h) – x]

=> (d/dx) e^2x = lim_x->0 [e^2x.e^2h – e^2x]/h [By Exponential Formula]

=> (d/dx) e^2x = lim_x->0 [e^2x.(e^2h – 1)]/h

=> (d/dx) e^2x = lim_x->0 e^2x.[(e^2h – 1)/h]

=> (d/dx) e^2x = e^2x.lim_h->0 [(e^2h – 1)/h]

=> (d/dx) e^2x = e^2x.2 [By Limit Formula]

=> (d/dx) e^2x = 2e^2x

Derivative of e^2x using Product Rule

To prove derivative of e^2x using Product Rule, we will use basic derivatives and exponential formulas which are listed below:

• e^{(x + y)} = e^x . e^y
• (d/dx) [u.v] = u’v + uv’

Let’s start proof of derivative of e^2x:

Let u = e^x and v = e^x, then e^2x = u×v

By Product Rule,

d/dx e^2x = (d/dx) [u×v]

=> (d/dx) e^2x = u’v + uv’

=> (d/dx) e^2x = (d/dx e^x).e^x + e^x.(d/dx e^x)

=> (d/dx) e^2x = e^x.e^x + e^x.e^x

=> (d/dx) e^2x = 2e^2x

Derivative of e^2x using Chain Rule

To prove derivative of e^2x using chain rule, we will use basic derivatives and exponential formulas which are listed below:

• (d/dx) e^x = e^x
• (d/dx) [f(g(x))] = f'(g(x)) . g'(x)

Let’s start proof of derivative of e^2x

We can write e^2x = f(g(x)) where f(x) = e^x and g(x) = 2x as a composite function

Then f'(x) = e^x and g'(x) = 2.

By chain rule, derivative of f(g(x)) is f'(g(x)).g'(x)

=> (d/dx) e^2x = f'(g(x)) . g'(x)

=> (d/dx) e^2x = f'(2x) . 2

=> (d/dx) e^2x = e^2x . 2

=> (d/dx) e^2x = 2e^2x

Derivative of e^2x by Logarithmic Differentiation

Logarithmic differentiation is used to differentiate exponential function and is thus used to find the derivative of e^2x.

let, y = e^2x

ln y = ln e^2x

ln y = 2x ln e (ln e = 1)

ln y = 2x

Differentiating both sides with respect to x,

(1/y) (dy/dx) = 2(1)

dy/dx = 2y

Substituting y = e^2x

d/dx(e^2x) = 2e^2x

nth Derivative of e^2x

If we differentiate e^2x n times nth derivative of e^2x is obtained,

1st derivative of e^2x = 2 e^2x

2nd derivative of e^2x = 4 e^2x

3rd derivative of e^2x = 8 e^2x

4th derivative of e^2x = 16 e^2x and so on.

Thus, n^th derivative of e^2x is given as,

dn/(dxⁿ) (e^2x) = 2n.e^2x

Examples on Derivative of e^2x

Some examples related to derivative of e^2x are,

Example 1: Find derivative of e^8x.

Solution:

Let y = e^8x

⇒ y’ = (d/dx) e^8x

Applying chain rule

y’ = (d/d 8x) (e^8x).(d/dx) (8x)

⇒ y’ = (e^8x).8

⇒ y’ = 8e^8x

Example 2: Evaluate derivative f(x) = (x³ + 5x² + 2x + 7)e^2x

Solution:

f(x) = (x³ + 5x² + 2x + 7).e^2x

⇒ f'(x) = (d /dx)[(x³ + 5x² + 2x + 7)e^2x]

Applying Product Rule

f'(x) = (d /dx)[(x³ + 5x² + 2x + 7)].e^2x + (x³ + 5x² + 2x + 7).(d /dx)[e^2x]

⇒ f'(x) = (3x² + 10x +2).e^2x + (x³ + 5x² + 2x + 7)(2e^2x)

⇒ f'(x) = (3x² + 10x +2)e^2x + 2.(x³ + 5x² + 2x + 7)e^2x

Example 3: Evaluate derivative e^5x+ x.e^2x

Solution:

Let z = e^5x + x.e^2x

Differentiating

z’ = (d/dx) [e^5x + x.e^2x]

⇒ z’ = (d/dx) e^5x + (d/dx)[x.e^2x]

Applying Chain Rule and Product Rule

z’ = 5.e^5x + (d/dx)(x).e^2x + x.(d/dx)(e^2x)

⇒ z’ = 5.e^5x + e^2x + x.(2e^2x)

Example 4: Find derivative of e^-x.

Solution:

= (d /dx) [e^-x]

= -e^-x

Example 5: Find derivative of e^2xcos x

Solution:

Applying Product Rule

= (d/dx) [e^2x cos x]

= e^2x d/dx (cos x) + cos x d/dx (e^2x)

= e^2x (-sin x) + cos x.(2e^2x)

Practice Questions on Derivative of e^2x

Some practice questions related to derivative of e^2x are,

Q1. Find the derivative of e^4x+1

Q2. Find the derivative of x².e^2x

Q3. Evaluate: (d/dx) [e^2x/(x² + 2)]

Q4. Evaluate the derivative of: e^2x. tan x

Q5. Find: [Tex]xe^{2^{x}}[/Tex]

FAQs on Derivative of e^2x

What is Derivative?

Derivative of a function is defined as the rate of change of the function with respect to a variable.

What is Formula for Derivative of e^2x.

Formula for the derivative of e^2x is: (d/dx) e^2x = 2e^2x

What are Different Methods to Prove Derivative of e^2x?

Different methods to prove derivative of e^2x are:

• By using First Principle of Derivative
• By Product Rule
• By Chain Rule

What is Derivative of 2.e^2x?

Derivative of 2.e^2x is 4e^2x.

Is derivative of e^2x same as integral of e^2x?

No, derivative of e^2x is not same as integral of e^2x

• Derivative of e^2x is 2e^2x.
• Integral of e^2x is e^2x / 2.

What is Derivative of e to the power x²?

Derivative of [Tex]e^{x^2}[/Tex] is [Tex]2x.e^{x^2}[/Tex]