Derivative of e^2x: Formula, Proof by First Principle, Chain Rule

What is the Derivative of e^2x? The derivative of e^2x is 2e^2x. Here, we will find the derivative of e^2x using the derivative of e^x. We will use three methods: substitution method, chain rule, and limit definition.

Derivative of e^2x Formula

The derivative of e^2x is 2e^2x. This can be written mathematically as follows: d/dx(e^2x) = 2 e^2x  or (e^2x)’ = 2 e^2x.

What is the derivative of e^2x?

At first, we will find the derivative of e^2x by the substitution method. This method is known as logarithmic differentiation. The following steps have to be followed in this method.

Step 1: Let

\[y=e^{2x}.\]

Step 2: Taking logarithms on both sides, we get that

\[\log_e y =\log_e e^{2x}\]

\[\Rightarrow \log_e y =2x\log_e e \]

\[\Rightarrow \log_e y =2x [\because \log_a a=1]\]

Step 3: Differentiating both sides of log_ey=2x with respect to x, we have

$\dfrac{1}{y} \dfrac{dy}{dx}=\dfrac{d}{dx}(2x)$

$\Rightarrow \dfrac{1}{y} \dfrac{dy}{dx}=2$

$\Rightarrow \dfrac{dy}{dx}=2y$

$\Rightarrow \dfrac{dy}{dx}=2e^{2x}$ $[\because y=e^{2x}]$

∴ The derivative of e^2x by the logarithmic differentiation is 2e^2x. That is, we have

\[\dfrac{d}{dx}(e^{2x})=2e^{2x}.\]

Derivative of e^2x using chain rule

Recall the chain rule of derivatives: $\frac{df}{dx}=\frac{df}{dz} \cdot \frac{dz}{dx}.$ To find the differentiation of e^2x by the chain rule, we put

z=2x

Now, $\dfrac{d}{dx}(e^{2x})=\dfrac{d}{dx}(e^z)$

$=\dfrac{d}{dz}(e^z) \cdot \dfrac{dz}{dx}$, by the chain rule.

$=\dfrac{d}{dz}(e^z) \cdot \dfrac{d}{dx}(2x)$

= e^z ⋅ 2 as (e^x)$’$=e^x

= 2e^2x  [∵ z=2x]

∴ the derivative of e^2x by chain rule is 2e^2x.

Also Read: 

Derivative of cube root of x: The derivative of the cube root of x is 1/(3x^{2/3}) Derivative of root x: The derivative of √x is 1/2√x Integration of root x: The integration of  √x is 2/3x^{3/2} Integration of mod x: The integration of  |x| is -x|x|/2 +c

Derivative of e^2x from first principle

From first principle of derivatives, we know the derivative of f(x) is given by

$f'(x)=\lim\limits_{h \to 0}\dfrac{f(x+h)-f(x)}{h}$

So taking f(x)=e^2x, the derivative of e^2x by first principle is

$(e^{2x})’= \lim\limits_{h \to 0} \dfrac{e^{2(x+h)}-e^{2x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{2x+2h}-e^{2x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{2x} \cdot e^{2h}-e^{2x}}{h}$

$=\lim\limits_{h \to 0} \dfrac{e^{2x}(e^{2h}-1)}{h}$

$=e^{2x}\lim\limits_{h \to 0} \Big(\dfrac{e^{2h}-1}{2h} \times 2 \Big)$

$=2e^{2x}\lim\limits_{h \to 0} \dfrac{e^{2h}-1}{2h}$

[Let $t=2h.$ Then $t \to 0$ as $x \to 0$]

$=2e^{2x}\lim\limits_{t \to 0} \dfrac{e^{t}-1}{t}$

$=2e^{2x} \cdot 1$

$=2e^{2x}$

This shows that the differentiation of e^2x by first principle is equal to 2e^2x.

n-th Derivative of e^2x

Differentiating e^2x with respect to x for n times, we will get the n^th derivative of e to the power x. The first order derivative of e^2x is 2e^2x. So the second order derivative of e^2x is equal to d/dx(2e^2x) = 2 d/dx(e^2x) = 2×2 e^2x = 2² e^2x. Now, we will understant the patterns of these derivatives:

• First derivative of e^2x is 2e^2x = 2¹ e^2x
• Second derivative of e^2x is 2² e^2x
• Third derivative of e^2x is 2³ e^2x  and so on.

Observing these patterns carefully, we get the n^th derivative of e^2x is 2ⁿ e^2x.

Important Notes on Derivative of e^2x:

• The differentiation of e^x is e^x but d/dx(e^2x) = 2e^2x.
• The derivative of e^-2x is -2e^-2x.
• In general, the derivative of e^mx is me^mx where m is a real number.

Application of Derivative of e^2x

From the above we know that

$\dfrac{d}{dx}(e^{2x})$ $=2e^{2x}$ $\cdots (I)$

Using this fact, we can calculate the derivatives of many exponential functions. We will find the derivative of $e^{2x+1}.$

Question 1: Find $\dfrac{d}{dx}(e^{2x+1})$

Solution:

Let z=2x+1.

$\dfrac{d}{dx}(e^{2x+1})$ $=\dfrac{d}{dx}(e^z)$

$=\dfrac{d}{dz}(e^z) \cdot \dfrac{dz}{dx}$ by the chain rule

$=e^z \cdot \dfrac{d}{dx}(2x+1)$ by Equation (I)

$=e^z \cdot 2$

$=2e^{2x+1}$  $[\because z=2x+1]$

Thus, the derivative of e^2x+1 is 2e^2x+1.

Question 2: What is the derivative of e²?

Solution:

Note that the function e² is a constant function of x. We know that the derivative of a constant function is 0. Thus, the derivative of e square is zero.

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