Proof of Derivative of e2x

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

In this article, we will learn in detail about the derivative of e to the power 2x, its formula, proof and examples based on it.

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.