Before walking to find the derivative that e2x, let united state recall a few facts about the exponential functions. In math, exponential functions are that the kind f(x) = ax, whereby 'a' is a consistent and 'x' is a variable. Here, the continuous 'a' have to be greater than 0 for f(x) to be an exponential function. Part other creates of exponential functions are abx, abkx, ex, pekx, etc. Thus, e2x is also an exponential duty and the derivative the e2x is 2e2x.
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We room going to discover the derivative of e2x in various methods and we will additionally solve a few examples making use of the same.
|1.||What is the Derivative the e^2x?|
|2.||Derivative that e^2x proof by an initial Principle|
|3.||Derivative of e^2x evidence by Chain Rule|
|4.||Derivative that e^2x evidence by Logarithmic Differentiation|
|5.||n^th Derivative the e^2x|
|6.||FAQs ~ above Derivative the e^2x|
What is the Derivative that e^2x?
The derivative that e2x through respect to x is 2e2x. We write this mathematically together 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 well-known as Euler's number and its worth is around 2.718) and the border formula the 'e' is lim ₙ→∞ (1 + (1/n))n. We deserve to do the differentiation of e2x in various methods such as:Using the first principleUsing the chain ruleUsing logarithmic differentiation
Derivative the e^2x Formula
The derivative of e2x is 2e2x. It can be written asd/dx (e2x) = 2e2x(or)(e2x)' = 2e2x
Let united state prove this in different methods as pointed out above.
Derivative the e^2x proof by an initial Principle
Here is the differentiation that e2x by the first principle. Because that this, let us assume that f(x) = e2x. Climate f(x + h) = e2(x + h) = e2x + 2h. Substituting these values in the formula of the derivative using first principle (which is likewise known as the limit definition of the derivative),
f'(x) = limₕ→₀
f'(x) = limₕ→₀
= e2x limₕ→₀ (e2h - 1) / h
Assume that 2h = t. Then together h → 0, 2h → 0. I.e., t → 0 as well. Climate the over limit becomes,
= e2x limₜ→₀ (et - 1) / (t / 2)
= 2e2x limₜ→₀ (et - 1) / t
Using limit formulas, we have actually limₜ→₀ (et - 1) / t = 1. So
f'(x) = 2e2x (1) = 2e2x
Thus, the derivative the e2x is found by the very first principle.
Derivative that e^2x evidence by Chain Rule
We have the right to do the differentiation the e2x making use of the chain rule since e2x deserve to be expressed as a composite function. I.e., we deserve to 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). Utilizing this,
d/dx (e2x) = f'(g(x)) · g'(x)
= f'(2x) · (2)
= e2x (2)
Thus, the derivative of e2x is found by utilizing the chain rule.
Derivative that e^2x evidence by Logarithmic Differentiation
We understand that the logarithmic differentiation is supplied to identify an exponential role and therefore it can be offered to discover the derivative of e2x. For this, let united state assume that y = e2x. As a process of logarithmic differentiation, us take the herbal logarithm (ln) on both political parties of the above equation. Then us get
ln y = ln e2x
One the the properties of logarithms is ln am = m ln a. Making use of this,
ln y = 2x ln e
We recognize that ln e = 1. So
ln y = 2x
Differentiating both sides with respect come x,
(1/y) (dy/dx) = 2(1)
dy/dx = 2y
Substituting y = e2x ago here,
d/dx(e2x) = 2e2x
Thus, we have uncovered the derivative the e2x by utilizing logarithmic differentiation.
n^th Derivative that e^2x
nthderivative of e2x is the derivative the e2x the is obtained by separating e2x repeatedly for n times. To find the nthderivative the e2xx, let us discover the first derivative, 2nd derivative, ... Approximately a couple of times to know the trend.1stderivative the e2x is 2e2x2ndderivative the e2x is 4e2x3rdderivative of e2x is 8e2x4thderivative that e2x is 16 e2xand for this reason on.
Thus, the nthderivative ofe2xis:dn/(dxn) (e2x) = 2ne2x
Important notes on Derivative of e2x:The derivative the e2x is NOT simply e2x, however it is 2e2x.In general, the derivative of eax is aeax.For example, the derivative that e-2x is -2e-2x, the derivative the e5x is 5e5x, etc.
Topics pertained to Derivative the e2x:
Examples using Derivative of e^2x
Example 1: find the derivative that e2x + 1.
Let f(x) = e2x + 1
Using the chain rule,
f'(x) = e2x + 1 d/dx (2x + 1)
= e2x + 1 (2)
= 2e2x + 1
Answer: The derivative the e2x + 1 is 2e2x + 1.
Example 2: What is the derivative the e2x + e-2x?
Let f(x) = e2x + e-2x
Using the chain rule,
f'(x) = e2x · d/dx (2x) + e-2x · d/dx (-2x)
= e2x (2) + e-2x (-2)
= 2 (e2x - e-2x)
Answer: The derivative the e2x + e-2x is 2 (e2x - e-2x).
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Practice concerns on Derivative of e^2x
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FAQs on Derivative of e^2x
What is Derivative the e2x?
The derivative of e2x is 2e2x. Mathematically, the is composed as d/dx(e2x) = 2e2x (or) (e2x)' = 2e2x.
How to differentiate e to the power of 2x?
Let f(x) = e2x. By using chain rule, the derivative that e2x is, e2x d/dx (2x) = e2x (2) = 2 e2x. Thus, the derivative of e to the power of 2x is 2e2x.
What is the Derivative of e3x?
Let f(x) = e3x. By applying chain rule, the derivative of e3x is, e3x d/dx (3x) = e3x (2) = 3 e3x. Thus, the derivative of e3x is 3e3x.
How to find the Derivative of e2x + 3?
Let us assume that f(x) = e2x + 3. Utilizing the chain rule, f'(x) = e2x + 3 d/dx (2x + 1) = e2x + 3 (2) = 2e2x + 3. Thus, the derivative the e2x + 3 is 2e2x + 3.
Is the Derivative that e2x exact same as the Integral the e2x?
No, the derivative the e2x is NOT very same as the integral of e2x.The derivative that e2x is 2e2x.The integral the e2x is e2x / 2.
What is the Derivative the e2x²?
Let f(x) = e2x². By the application of chain rule, f'(x) = e2x² d/dx (2x2) = e2x² (4x) = 4x e2x². Thus, the derivative of e2x² is 4x e2x².
How to find the Derivative the e2x by an initial Principle?
Let f(x) = e2x. By very first principle, f'(x) = limₕ→₀
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What is the Derivative that e2x sin 3x?
Let f(x) = e2x sin 3x. Through product rule, f'(x) = e2x d/dx (sin 3x) + sin 3x d/dx (e2x) = e2x (cos 3x) d/dx (3x) + sin 3x (2e2x) = e2x (3 cos 3x + 2 sin 3x). Thus, the derivative that e2x sin 3x is e2x (3 cos 3x + 2 sin 3x).