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Derivative of e^2x
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^{2}^{x}
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^{th} derivative of e^{2x} is the derivative of e^{2x} that is obtained by differentiating e^{2x} repeatedly for n times. To find the n^{th} derivative of e^{2x}x, let us find the first derivative, second derivative, ... up to a few times to understand the trend.
 1^{st} derivative of e^{2x} is 2 e^{2x}
 2^{nd} derivative of e^{2x} is 4 e^{2x}
 3^{rd} derivative of e^{2x} is 8 e^{2x}
 4^{th} derivative of e^{2x} is 16 e^{2x }and so on.
Thus, the n^{th} derivative of e^{2x} is:
 d^{n}/(dx^{n}) (e^{2x}) = 2^{n} e^{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:
Examples Using Derivative of e^2x

Example 1: Find the derivative of e^{2x + 1}.
Solution:
Let f(x) = e^{2x + 1}
Using the chain rule,
f'(x) = e^{2x + 1} d/dx (2x + 1)
= e^{2x + 1} (2)
= 2e^{2x + 1}
Answer: The derivative of e^{2x + 1} is 2e^{2x + 1}.

Example 2: What is the derivative of e^{2x} + e^{2x}?
Solution:
Let f(x) = e^{2x} + e^{2x}
Using the chain rule,
f'(x) = e^{2x} · d/dx (2x) + e^{2x} · d/dx (2x)
= e^{2x} (2) + e^{2x} (2)
= 2 (e^{2x}  e^{2x})
Answer: The derivative of e^{2x} + e^{2x} is 2 (e^{2x}  e^{2x}).

Example 3: What is the derivative of e^{2x} sin x.
Solution:
Let f(x) = e^{2x} sin x.
Using the product rule,
f'(x) = e^{2x} d/dx (sin x) + sin x d/dx (e^{2x})
= e^{2x} (cos x) + sin x (2e^{2x})
= e^{2x} (cos x + 2 sin x)
Answer: The derivative of e^{2x} sin x is e^{2x} (cos x + 2 sin x).
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^{2x}^{²}?
Let f(x) = e^{2x}^{²}. By the application of chain rule, f'(x) = e^{2x}^{² }d/dx (2x^{2}) = e^{2x}^{²} (4x) = 4x e^{2x}^{²}. Thus, the derivative of e^{2x}^{²} is 4x e^{2x}^{²}.
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).
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