Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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2:53 minutes

Problem 3.6.77

Textbook Question

Find ds/dt when θ = 3π/2 if s = cosθ and dθ/dt = 5.

Verified step by step guidance

Identify the given function: s = cos(θ). We need to find ds/dt, the rate of change of s with respect to time.

Use the chain rule for differentiation, which states that ds/dt = (ds/dθ) * (dθ/dt).

Differentiate s = cos(θ) with respect to θ to find ds/dθ. The derivative of cos(θ) is -sin(θ), so ds/dθ = -sin(θ).

Substitute the given value of dθ/dt = 5 into the chain rule expression: ds/dt = -sin(θ) * 5.

Evaluate -sin(θ) at θ = 3π/2. Since sin(3π/2) = -1, substitute this value into the expression to find ds/dt.

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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 concept in calculus used to differentiate composite functions. It states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x. In this problem, it helps find ds/dt by relating ds/dθ and dθ/dt.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial for solving calculus problems involving these functions. The derivative of cos(θ) with respect to θ is -sin(θ). This knowledge is essential for finding ds/dθ when s = cos(θ), which is a step in applying the chain rule to find ds/dt.

Evaluating Trigonometric Functions

Evaluating trigonometric functions at specific angles is necessary for solving problems involving these functions. At θ = 3π/2, the value of sin(θ) is -1. This evaluation is crucial for determining the value of ds/dθ at the given angle, which is then used to find ds/dt using the chain rule.

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