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Calculate the derivative of the function

g(x) = e^{2x}

Jan 3, 2024

Calculate the derivative of the function

g(x) = e^{2x}

Solution by Steps

step 1

To calculate the derivative of the function

g(x) = e^{2x}

, we use the chain rule for differentiation. The chain rule states that if a function

y

is composed of a function

u

inside another function

v

, where

y = v(u(x))

, then the derivative of

y

with respect to

x

\frac{dy}{dx} = \frac{dv}{du} \cdot \frac{du}{dx}

step 2

Let

u = 2x

and

v = e^u

. Then

g(x) = v(u(x))

. We need to find

\frac{dv}{du}

and

\frac{du}{dx}

step 3

The derivative of

u

with respect to

x

\frac{du}{dx} = 2

step 4

The derivative of

v

with respect to

u

\frac{dv}{du} = e^u

because the derivative of

e^u

with respect to

u

e^u

step 5

Applying the chain rule, the derivative of

g(x)

with respect to

x

\frac{dg}{dx} = \frac{dv}{du} \cdot \frac{du}{dx} = e^u \cdot 2

step 6

Substituting

u = 2x

back into the expression, we get

\frac{dg}{dx} = e^{2x} \cdot 2

Answer

The derivative of the function

g(x) = e^{2x}

2e^{2x}

Key Concept

Chain Rule for Differentiation

Explanation

The chain rule is used to differentiate composite functions, and in this case, it is applied to the exponential function with a linear exponent.