Derivative Of E^2x - Formula, Proof, Examples - Cuemath

Derivative of e^2x

Before going to find the derivative of e2x, let us recall a few facts about the exponential functions. In math, exponential functions are of the form f(x) = ax, 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 abx, abkx, ex, pekx, etc. Thus, e2x is also an exponential function and the derivative of e2x is 2e2x.

We are going to find the derivative of e2x in different methods and we will also solve a few examples using the same.

1.	What is the Derivative of e^2x?
2.	Derivative of e^2x Proof by First Principle
3.	Derivative of e^2x Proof by Chain Rule
4.	Derivative of e^2x Proof by Logarithmic Differentiation
5.	n^th Derivative of e^2x
6.	FAQs on Derivative of e^2x

What is the Derivative of e^2x?

The derivative of e2x with respect to x is 2e2x. We write this mathematically as d/dx (e2x) = 2e2x (or) (e2x)' = 2e2x. Here, f(x) = e2x 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 e2x in different methods such as:

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

Derivative of e^2x Formula

The derivative of e2x is 2e2x. It can be written as

• d/dx (e2x) = 2e2x (or)
• (e2x)' = 2e2x

Let us prove this in different methods as mentioned above.

Derivative of e^2x Proof by First Principle

Here is the differentiation of e2x by the first principle. For this, let us assume that f(x) = e2x. Then f(x + h) = e2(x + h) = e2x + 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 [e2x + 2h - e2x] / h

= limₕ→₀ [e2x e2h - e2x] / h

= limₕ→₀ [e2x (e2h - 1) ] / h

= e2x limₕ→₀ (e2h - 1) / h

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

= e2x limₜ→₀ (et - 1) / (t / 2)

= 2e2x limₜ→₀ (et - 1) / t

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

f'(x) = 2e2x (1) = 2e2x

Thus, the derivative of e2x is found by the first principle.

Derivative of e^2x Proof by Chain Rule

We can do the differentiation of e2x using the chain rule because e2x can be expressed as a composite function. i.e., we can write e2x = f(g(x)) where f(x) = ex and g(x) = 2x (one can easily verify that f(g(x)) = e2x).

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

d/dx (e2x) = f'(g(x)) · g'(x)

= f'(2x) · (2)

= e2x (2)

= 2e2x

Thus, the derivative of e2x 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 e2x. For this, let us assume that y = e2x. 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 e2x

One of the properties of logarithms is ln am = 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 = e2x back here,

d/dx(e2x) = 2e2x

Thus, we have found the derivative of e2x by using logarithmic differentiation.

n^th Derivative of e^2x

nth derivative of e2x is the derivative of e2x that is obtained by differentiating e2x repeatedly for n times. To find the nth derivative of e2xx, let us find the first derivative, second derivative, ... up to a few times to understand the trend.

• 1st derivative of e2x is 2 e2x
• 2nd derivative of e2x is 4 e2x
• 3rd derivative of e2x is 8 e2x
• 4th derivative of e2x is 16 e2x and so on.

Thus, the nth derivative of e2x is:

• dn/(dxn) (e2x) = 2n e2x

Important Notes on Derivative of e2x:

• The derivative of e2x is NOT just e2x, but it is 2e2x.
• In general, the derivative of eax is aeax. For example, the derivative of e-2x is -2e-2x, the derivative of e5x is 5e5x, etc.

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