Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.13

Chapter 3, Problem 3.13

Find the derivatives of the functions in Exercises 1–42.

s = cos⁴ (1 - 2t)

Verified step by step guidance

Identify the function: We have s = cos⁴(1 - 2t). This is a composite function where the outer function is u⁴ and the inner function is u = cos(1 - 2t).

Apply the chain rule: The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). Here, we need to find the derivative of the outer function with respect to the inner function, and then multiply it by the derivative of the inner function with respect to t.

Differentiate the outer function: The outer function is u⁴. The derivative of u⁴ with respect to u is 4u³. So, we have 4(cos(1 - 2t))³.

Differentiate the inner function: The inner function is cos(1 - 2t). The derivative of cos(x) is -sin(x), and using the chain rule again, the derivative of (1 - 2t) with respect to t is -2. Therefore, the derivative of cos(1 - 2t) is -sin(1 - 2t) * (-2) = 2sin(1 - 2t).

Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function to get the final derivative: 4(cos(1 - 2t))³ * 2sin(1 - 2t).

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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 principle in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and v, such that y = f(g(x)), then the derivative of y with respect to x is the product of the derivative of f with respect to g and the derivative of g with respect to x. This rule is essential for finding derivatives of functions like s = cos⁴(1 - 2t), where multiple layers of functions are involved.

Power Rule

The Power Rule is a basic differentiation rule that states if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives of polynomial and power functions. In the context of the given function s = cos⁴(1 - 2t), applying the Power Rule will help differentiate the outer function raised to the fourth power.

Trigonometric Derivatives

Trigonometric derivatives refer to the derivatives of trigonometric functions, which are essential for solving problems involving angles and periodic functions. For example, the derivative of cos(x) is -sin(x). In the function s = cos⁴(1 - 2t), understanding the derivative of the cosine function is crucial for applying the Chain Rule and finding the overall derivative of the function.

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Derivatives of Other Inverse Trigonometric Functions

Related Practice

Textbook Question

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

f(x) = { 2x + tan x, x ≥ 0

x², x < 0

Textbook Question

The diameter of a sphere is measured as 100 ± 1 cm and the volume is calculated from this measurement. Estimate the percentage error in the volume calculation.

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

𝔂 = 3 .

(5x² + sin 2x)³/²

Textbook Question

Linearization for Approximation

In Exercises 7–12, find a linearization at a suitably chosen integer near a at which the given function and its derivative are easy to evaluate.

f(x) = ∛x, a = 8.5

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

s = √ t .

1 + √ t