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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.6.21
Chapter 3, Problem 3.6.21

Find the derivatives of the functions in Exercises 19–40.


s = (4 / 3π)sin(3t) + (4 / 5π)cos(5t)

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Identify the function components: The function s(t) is composed of two trigonometric terms: (4 / 3π)sin(3t) and (4 / 5π)cos(5t). Each term will be differentiated separately.
Apply the derivative rule for sine: The derivative of sin(kt) with respect to t is k * cos(kt). For the term (4 / 3π)sin(3t), apply this rule to find its derivative.
Differentiate the first term: The derivative of (4 / 3π)sin(3t) is (4 / 3π) * 3 * cos(3t). Simplify this expression to get the derivative of the first term.
Apply the derivative rule for cosine: The derivative of cos(kt) with respect to t is -k * sin(kt). For the term (4 / 5π)cos(5t), apply this rule to find its derivative.
Differentiate the second term: The derivative of (4 / 5π)cos(5t) is -(4 / 5π) * 5 * sin(5t). Simplify this expression to get the derivative of the second term.

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Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and concavity.
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Derivatives

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. They are periodic functions, meaning they repeat their values in regular intervals. Understanding their derivatives is essential, as they follow specific rules: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x), which are crucial for differentiating functions involving these terms.
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Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with trigonometric functions that are themselves functions of another variable, such as t in the given expression.
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Related Practice
Textbook Question

Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the lateral surface area S = πr√(r² + h²) of a right circular cone when the radius changes from r₀ to r₀ + dr and the height does not change

Textbook Question

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.


(y - x)² = 2x + 4, (6, 2)

Textbook Question

a. Graph the function


ƒ(x) = { x², -1 ≤ x < 0

{ -x², 0 ≤ x ≤ 1.


b. Is ƒ continuous at x = 0?

c. Is ƒ differentiable at x = 0?


Give reasons for your answers.

Textbook Question

Find the points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent line is


a. perpendicular to the line y = 1 - (x/24).

_

b. parallel to the line y = √2 - 12x.

Textbook Question

In Exercises 41–58, find dy/dt.


y = tan²(sin³(t))

Textbook Question

Differential Estimates of Change


In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.


The change in the surface area S = 6x² of a cube when the edge lengths change from x₀ to x₀ + dx