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Calculate the derivative of the function

g(x) = e^{2x}

Jan 22, 2024

Calculate the derivative of the function

g(x) = e^{2x}

Generated Graph

Solution by Steps

step 1

To find the derivative of the function

g(x) = e^{2x}

, we apply the chain rule

step 2

The derivative of

e^{u}

with respect to

u

e^{u}

, where

u

is a function of

x

step 3

In our case,

u = 2x

. The derivative of

u

with respect to

x

\frac{du}{dx} = 2

step 4

Applying the chain rule, the derivative of

g(x)

\frac{d}{dx}(e^{2x}) = \frac{d}{du}(e^{u}) \cdot \frac{du}{dx}

step 5

Substituting

u = 2x

and

\frac{du}{dx} = 2

, we get

\frac{d}{dx}(e^{2x}) = e^{2x} \cdot 2

step 6

Therefore, the derivative of

g(x)

2e^{2x}

Answer

g'(x) = 2e^{2x}

Key Concept

Chain Rule in Differentiation

Explanation

The chain rule is used to differentiate composite functions. In this case,

e^{2x}

is a composite function where

e^{u}

is the outer function and

u = 2x

is the inner function. The derivative is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function with respect to

x