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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.9.55
Chapter 3, Problem 3.9.55

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.
f (x) = (4 sin x+2)^cos x; a = π

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First, recognize that the function f(x) = (4 sin x + 2)^cos x is a tower function, which can be expressed in the form g(x)^h(x). To differentiate this type of function, use the logarithmic differentiation method.
Take the natural logarithm of both sides: ln(f(x)) = ln((4 sin x + 2)^cos x). This simplifies to ln(f(x)) = cos x * ln(4 sin x + 2) using the property of logarithms ln(a^b) = b * ln(a).
Differentiate both sides with respect to x. For the left side, use the chain rule: d/dx[ln(f(x))] = (1/f(x)) * f'(x). For the right side, apply the product rule: d/dx[cos x * ln(4 sin x + 2)] = -sin x * ln(4 sin x + 2) + cos x * (1/(4 sin x + 2)) * (4 cos x).
Solve for f'(x) by multiplying both sides by f(x): f'(x) = f(x) * [-sin x * ln(4 sin x + 2) + cos x * (4 cos x)/(4 sin x + 2)].
Finally, evaluate f'(x) at x = π. Substitute x = π into the expression for f'(x) and simplify. Remember that sin(π) = 0 and cos(π) = -1, which will help simplify the calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The Chain Rule is a fundamental differentiation technique used to find the derivative of composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for differentiating functions like f(x) = (4 sin x + 2)^(cos x), where both the base and the exponent are functions of x.
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Intro to the Chain Rule

Product Rule

The Product Rule is used to differentiate products of two functions. If u(x) and v(x) are two differentiable functions, the derivative of their product is given by u'v + uv'. This rule is particularly relevant in the context of the given function, as it involves differentiating the base and the exponent, both of which are products of functions themselves.
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The Product Rule

Exponential Functions and Logarithmic Differentiation

Exponential functions, particularly those in the form of g^h, where both g and h are functions of x, can be differentiated using logarithmic differentiation. This technique involves taking the natural logarithm of both sides, simplifying the expression, and then differentiating. This method is useful for the function f(x) = (4 sin x + 2)^(cos x) because it allows for easier manipulation of the exponent and base during differentiation.
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Logarithmic Differentiation
Related Practice
Textbook Question

15–48. Derivatives Find the derivative of the following functions.

f(x) = 2^x/2^x+1

Textbook Question

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

y = xe^y

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23–51. Calculating derivatives Find the derivative of the following functions.

y = sin x / 1 + cos x

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = 2e^x-1 / 2e^x+1

Textbook Question

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

Suppose the speed of a car approaching a stop sign is given by v (t) = (t-5)², for 0 ≤ t ≤ 5, where t is measured in seconds and v(t) is measured in meters per second. Find v′(3).

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

g(x) = 6x⁵ - 5/2 x² + x + 5

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