Question

Find $f^{\prime}(x)$ if $f(x)$ is the given expression. $$ \ln \left(e^{4 x}+9\right) $$

   Find $f^{\prime}(x)$ if $f(x)$ is the given expression.
$$
\ln \left(e^{4 x}+9\right)
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 7, Problem 17

Instant Answer

Solved by Expert Steven Clarke

03/04/2022

Step 1

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, our outer function is $\ln(x)$ and our inner function is $e^{4x}+9$. Show more…

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Find $f^{\prime}(x)$ if $f(x)$ is the given expression. $$ \ln \left(e^{4 x}+9\right) $$

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Key Concepts

Chain Rule

The chain rule is a technique for differentiating composite functions. It involves taking the derivative of the outer function while keeping the inner function unchanged, and then multiplying by the derivative of the inner function. This rule is critical when dealing with functions that are nested within one another.

Differentiation of Logarithmic Functions

Logarithmic differentiation leverages the fact that the derivative of the natural logarithm ln(g(x)) is given by g'(x)/g(x). This principle simplifies the differentiation process for functions where the logarithm is applied to a variable expression.

Differentiation of Exponential Functions

Exponential functions, such as those involving e raised to a power of a function, have a specific rule for differentiation: the derivative of e^(k*x) is k*e^(k*x). This concept is fundamental when exponential expressions appear within more complex functions.

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