Question

Find $f^{\prime}(x)$ if $f(x)$ equals the given expression. $$ e^{\sin 5 x} $$

   Find $f^{\prime}(x)$ if $f(x)$ equals the given expression.
$$
e^{\sin 5 x}
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 7, Problem 22

Instant Answer

Solved by Expert Gregory Higby

03/01/2022

Step 1

In this case, the outer function is $e^x$ and the inner function is $\sin 5x$. Show more…

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Find $f^{\prime}(x)$ if $f(x)$ equals the given expression. $$ e^{\sin 5 x} $$

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Key Concepts

Trigonometric Function Derivative

Trigonometric functions, such as sine and cosine, have well-established derivatives. For example, the derivative of sin(x) is cos(x). When the argument of a trigonometric function is itself a function of x, like sin(5x), the chain rule is applied to account for the derivative of the inner function, resulting in a factor from the derivative of 5x. This rule is essential for correctly differentiating trigonometric expressions that are composed with linear or other functions.

Chain Rule

The chain rule is a fundamental principle in calculus used for differentiating composite functions. It states that if a function y = f(g(x)) is composed of an outer function f and an inner function g, then the derivative is f'(g(x)) multiplied by g'(x). This rule is key when dealing with functions that have multiple layers of composition.

Exponential Function Derivative

The derivative of an exponential function where the base is the number e is unique because e raised to any function has a derivative that is the same exponential function times the derivative of the exponent. Specifically, if f(x) = e^(u(x)), then its derivative is f'(x) = e^(u(x)) * u'(x). This simplicity makes exponential functions elegant and frequently encountered in calculus.

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