Derivative of e^2x - Formula, Proof, Examples (2024)

Before going to find the derivative of e^2x, let us recall a few facts about the exponential functions. In math, exponential functions are of the form f(x) = a^x, where 'a' is a constant and 'x' is a variable. Here, the constant 'a' should be greater than 0 for f(x) to be an exponential function. Some other forms of exponential functions are ab^x, ab^kx, e^x, pe^kx, etc. Thus, e^2x is also an exponential function and the derivative of e^2x is 2e^2x.

We are going to find the derivative of e^2x in different methods and we will also solve a few examples using the same.

What is the Derivative of e^2x?

The derivative of e^2x with respect to x is 2e^2x. We write this mathematically as d/dx (e^2x) = 2e^2x (or) (e^2x)' = 2e^2x. Here, f(x) = e^2x is an exponential function as the base is 'e' is a constant (which is known as Euler's number and its value is approximately 2.718) and the limit formula of 'e' is lim ₙ→∞ (1 + (1/n))ⁿ. We can do the differentiation of e^2x in different methods such as:

• Using the first principle
• Using the chain rule
• Using logarithmic differentiation

Derivative of e^2x Formula

The derivative of e^2x is 2e^2x. It can be written as

• d/dx (e^2x) = 2e^2x(or)
• (e^2x)' = 2e^2x

Let us prove this in different methods as mentioned above.

Derivative of e^2x Proof by First Principle

Here is the differentiation of e^2x by the first principle. For this, let us assume that f(x) = e^2x. Then f(x + h) = e^{2(x + h)} = e^{2x + 2h}. Substituting these values in the formula of the derivative using first principle (which is also known as the limit definition of the derivative),

f'(x) = limₕ→₀ [f(x + h) - f(x)] / h

f'(x) = limₕ→₀ [e^{2x + 2h} - e^2x] / h

= limₕ→₀ [e^2x e^2h - e^2x] / h

= limₕ→₀ [e^2x (e^2h - 1) ] / h

= e^2x limₕ→₀ (e^2h - 1) / h

Assume that 2h = t. Then as h → 0, 2h → 0. i.e., t → 0 as well. Then the above limit becomes,

= e^2x limₜ→₀ (e^t - 1) / (t / 2)

= 2e^2x limₜ→₀ (e^t - 1) / t

Using limit formulas, we have limₜ→₀ (e^t - 1) / t = 1. So

f'(x) = 2e^2x (1) = 2e^2x

Thus, the derivative of e^2x is found by the first principle.

Derivative of e^2x Proof by Chain Rule

We can do the differentiation of e^2x using the chain rule because e^2x can be expressed as a composite function. i.e., we can write e^2x = f(g(x)) where f(x) = e^x and g(x) = 2x (one can easily verify that f(g(x)) = e^2x).

Then f'(x) = e^x and g'(x) = 2. By chain rule, the derivative of f(g(x)) is f'(g(x)) · g'(x). Using this,

d/dx (e^2x) = f'(g(x)) · g'(x)

= f'(2x) · (2)

= e^2x (2)

= 2e^2x

Thus, the derivative of e^2x is found by using the chain rule.

Derivative of e^2x Proof by Logarithmic Differentiation

We know that the logarithmic differentiation is used to differentiate an exponential function and hence it can be used to find the derivative of e^2x. For this, let us assume that y = e^2x. As a process of logarithmic differentiation, we take the natural logarithm (ln) on both sides of the above equation. Then we get

ln y = ln e^2x

One of the properties of logarithms is ln a^m = m ln a. Using this,

ln y = 2x ln e

We know that ln e = 1. So

ln y = 2x

Differentiating both sides with respect to x,

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

dy/dx = 2y

Substituting y = e^2x back here,

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

Thus, we have found the derivative of e^2x by using logarithmic differentiation.

n^th Derivative of e^2x

n^thderivative of e^2x is the derivative of e^2x that is obtained by differentiating e^2x repeatedly for n times. To find the n^thderivative of e^2xx, let us find the first derivative, second derivative, ... up to a few times to understand the trend.

• 1^stderivative of e^2x is 2e^2x
• 2^ndderivative of e^2x is 4e^2x
• 3^rdderivative of e^2x is 8e^2x
• 4^thderivative of e^2x is 16 e^2xand so on.

Thus, the n^thderivative ofe^2xis:

• dⁿ/(dxⁿ) (e^2x) = 2ⁿe^2x

Important Notes on Derivative of e^2x:

• The derivative of e^2x is NOT just e^2x, but it is 2e^2x.
• In general, the derivative of e^ax is ae^ax.
  For example, the derivative of e^-2x is -2e^-2x, the derivative of e^5x is 5e^5x, etc.

☛Related Topics:

• Derivative of log x
• Derivative of ln x
• Derivative Calculator
• Calculus Calculator

FAQs on Derivative of e^2x

What is the Formula of Derivative of e^2x?

The derivative of e^2x is 2e^2x. Mathematically, it is written as d/dx(e^2x) = 2e^2x (or) (e^2x)' = 2e^2x.

How to Differentiate e to the Power of 2x?

Let f(x) = e^2x. By applying chain rule, the derivative of e^2x is, e^2x d/dx (2x) = e^2x (2) = 2 e^2x. Thus, the derivative of e to the power of 2x is 2e^2x.

What is the Derivative of e^3x?

Let f(x) = e^3x. By applying chain rule, the derivative of e^3x is, e^3x d/dx (3x) = e^3x (2) = 3 e^3x. Thus, the derivative of e^3x is 3e^3x.

How to Find the Derivative of e^{2x + 3}?

Let us assume that f(x) = e^{2x + 3}. Using the chain rule, f'(x) = e^{2x + 3} d/dx (2x + 1) = e^{2x + 3} (2) = 2e^{2x + 3}. Thus, the derivative of e^{2x + 3} is 2e^{2x + 3}.

Is the Derivative of e^2x Same as the Integral of e^2x?

No, the derivative of e^2x is NOT the same as the integral of e^2x.

• The derivative of e^2x is 2e^2x.
• The integral of e^2x is e^2x / 2.

What is the Derivative of e^2x²?

Let f(x) = e^2x². By the application of chain rule, f'(x) = e^2x² d/dx (2x²) = e^2x² (4x) = 4x e^2x². Thus, the derivative of e^2x² is 4x e^2x².

How to Find the Derivative of e^2x by First Principle?

Let f(x) = e^2x. By first principle, f'(x) = limₕ→₀ [f(x + h) - f(x)] / h = limₕ→₀ [e^{2x + 2h} - e^2x] / h = limₕ→₀ [e^2x e^2h - e^2x] / h = limₕ→₀ [e^2x (e^2h - 1) ] / h = 2e^2x (1) = 2e^2x. Thus, the derivative of e^2x by first principle is 2e^2x.

What is the Derivative of e^2x sin 3x?

Let f(x) = e^2x sin 3x. By product rule, f'(x) = e^2x d/dx (sin 3x) + sin 3x d/dx (e^2x) = e^2x (cos 3x) d/dx (3x) + sin 3x (2e^2x) = e^2x (3 cos 3x + 2 sin 3x). Thus, the derivative of e^2x sin 3x is e^2x (3 cos 3x + 2 sin 3x).