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Chapter 3: Problem 13

Differentiate. $$g(x)=e^{3 x}$$

Short Answer

Expert verified
The derivative is \( g'(x) = 3e^{3x} \).

Step by step solution

01

Identify the function

The given function is \( g(x) = e^{3x} \), where the base of the exponential function is Euler's number \(e\).
02

Apply the chain rule

To differentiate \( g(x) = e^{3x} \), use the chain rule. According to the chain rule, if \( y = e^{u} \) and \( u = 3x \), then \( \frac{dy}{dx} = e^{u} \frac{du}{dx} \).
03

Differentiate the exponent

First, find \( \frac{du}{dx} \). Here, \( u = 3x \), so \( \frac{du}{dx} = 3 \).
04

Combine the results

Now substitute back into the chain rule formula: \( \frac{d}{dx}[e^{3x}] = e^{3x} \times 3 \).
05

Provide the final answer

Thus, the derivative of \( g(x) = e^{3x} \) is \( g'(x) = 3e^{3x} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

exponential function
The exponential function is one of the most important functions in mathematics. It is often written as \( e^x ot...\text{where} \baz=text {sp;}]e \ isthe-baset'sapproximately}2.71828 \). Exponential functions grow rapidly and curve upward or downward depending on whether the exponent is positive or negative.

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