Derivative of e^2x
Derivative of e2x is 2e2x. Derivative of e2x refers to the process of finding the change in the exponential function e2x with respect to the independent variable x. e2x 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 e2x and its formula including the proof of the formula using the first principle of derivatives, product rule, and chain rule in detail.
Table of Content
- What is Derivative in Math?
- What is Derivative of e2x?
- Proof of Derivative of e2x
- Derivative of e2x by Logarithmic Differentiation
- nth Derivative of e2x
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 e2x?
Among exponential derivatives, the derivative of e2x is one of the derivatives. Derivative of e2x is 2e2x. The derivative of e2x is the rate of change to the variable i.e., x. The resultant of the derivative of e2x is 2e2x. Mathematically derivative of e2x is represented as, d/dx(e2x) = 2e2x
Derivative of e2x Formula
Formula for derivative of e2x is given by:
(d/dx) [e2x] = 2e2x
(e2x)’ = 2e2x
Proof of Derivative of e2x
Derivative of e2x can be proved using the following ways:
- By using First Principle of Derivative
- By using Product Rule
- By using Chain Rule
Derivative of e2x using First Principle of Derivatives
To prove derivative of e2x using First Principle of Derivatives, we will use basic limits and exponential formulas which are listed below:
- e(x + y) = ex.ey
- limx->0 (ex – 1)/x = 1
Let’s start proof for derivative of e2x:
By First Principle of Derivatives,
(d/dx) e2x = limx->0 [e(2*(x + h)) – e(2x)]/[(x + h) – x]
=> (d/dx) e2x = limx->0 [e2x.e2h – e2x]/h [By Exponential Formula]
=> (d/dx) e2x = limx->0 [e2x.(e2h – 1)]/h
=> (d/dx) e2x = limx->0 e2x.[(e2h – 1)/h]
=> (d/dx) e2x = e2x.limh->0 [(e2h – 1)/h]
=> (d/dx) e2x = e2x.2 [By Limit Formula]
=> (d/dx) e2x = 2e2x
Derivative of e2x using Product Rule
To prove derivative of e2x using Product Rule, we will use basic derivatives and exponential formulas which are listed below:
- e(x + y) = ex . ey
- (d/dx) [u.v] = u’v + uv’
Let’s start proof of derivative of e2x:
Let u = ex and v = ex, then e2x = u×v
By Product Rule,
d/dx e2x = (d/dx) [u×v]
=> (d/dx) e2x = u’v + uv’
=> (d/dx) e2x = (d/dx ex).ex + ex.(d/dx ex)
=> (d/dx) e2x = ex.ex + ex.ex
=> (d/dx) e2x = 2e2x
Derivative of e2x using Chain Rule
To prove derivative of e2x using chain rule, we will use basic derivatives and exponential formulas which are listed below:
- (d/dx) ex = ex
- (d/dx) [f(g(x))] = f'(g(x)) . g'(x)
Let’s start proof of derivative of e2x
We can write e2x = f(g(x)) where f(x) = ex and g(x) = 2x as a composite function
Then f'(x) = ex and g'(x) = 2.
By chain rule, derivative of f(g(x)) is f'(g(x)).g'(x)
=> (d/dx) e2x = f'(g(x)) . g'(x)
=> (d/dx) e2x = f'(2x) . 2
=> (d/dx) e2x = e2x . 2
=> (d/dx) e2x = 2e2x
Derivative of e2x by Logarithmic Differentiation
Logarithmic differentiation is used to differentiate exponential function and is thus used to find the derivative of e2x.
let, y = e2x
ln y = ln e2x
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 = e2x
d/dx(e2x) = 2e2x
nth Derivative of e2x
If we differentiate e2x n times nth derivative of e2x is obtained,
1st derivative of e2x = 2 e2x
2nd derivative of e2x = 4 e2x
3rd derivative of e2x = 8 e2x
4th derivative of e2x = 16 e2x and so on.
Thus, nth derivative of e2x is given as,
dn/(dxn) (e2x) = 2n.e2x
Also, Check
Examples on Derivative of e2x
Some examples related to derivative of e2x are,
Example 1: Find derivative of e8x.
Solution:
Let y = e8x
⇒ y’ = (d/dx) e8x
Applying chain rule
y’ = (d/d 8x) (e8x).(d/dx) (8x)
⇒ y’ = (e8x).8
⇒ y’ = 8e8x
Example 2: Evaluate derivative f(x) = (x3 + 5x2 + 2x + 7)e2x
Solution:
f(x) = (x3 + 5x2 + 2x + 7).e2x
⇒ f'(x) = (d /dx)[(x3 + 5x2 + 2x + 7)e2x]
Applying Product Rule
f'(x) = (d /dx)[(x3 + 5x2 + 2x + 7)].e2x + (x3 + 5x2 + 2x + 7).(d /dx)[e2x]
⇒ f'(x) = (3x2 + 10x +2).e2x + (x3 + 5x2 + 2x + 7)(2e2x)
⇒ f'(x) = (3x2 + 10x +2)e2x + 2.(x3 + 5x2 + 2x + 7)e2x
Example 3: Evaluate derivative e5x+ x.e2x
Solution:
Let z = e5x + x.e2x
Differentiating
z’ = (d/dx) [e5x + x.e2x]
⇒ z’ = (d/dx) e5x + (d/dx)[x.e2x]
Applying Chain Rule and Product Rule
z’ = 5.e5x + (d/dx)(x).e2x + x.(d/dx)(e2x)
⇒ z’ = 5.e5x + e2x + x.(2e2x)
Example 4: Find derivative of e-x.
Solution:
= (d /dx) [e-x]
= -e-x
Example 5: Find derivative of e2xcos x
Solution:
Applying Product Rule
= (d/dx) [e2x cos x]
= e2x d/dx (cos x) + cos x d/dx (e2x)
= e2x (-sin x) + cos x.(2e2x)
Practice Questions on Derivative of e2x
Some practice questions related to derivative of e2x are,
Q1. Find the derivative of e4x+1
Q2. Find the derivative of x2.e2x
Q3. Evaluate: (d/dx) [e2x/(x2 + 2)]
Q4. Evaluate the derivative of: e2x. tan x
Q5. Find:
FAQs on Derivative of e2x
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 e2x.
Formula for the derivative of e2x is: (d/dx) e2x = 2e2x
What are Different Methods to Prove Derivative of e2x?
Different methods to prove derivative of e2x are:
- By using First Principle of Derivative
- By Product Rule
- By Chain Rule
What is Derivative of 2.e2x?
Derivative of 2.e2x is 4e2x.
Is derivative of e2x same as integral of e2x?
No, derivative of e2x is not same as integral of e2x
- Derivative of e2x is 2e2x.
- Integral of e2x is e2x / 2.
What is Derivative of e to the power x2?
Derivative of is