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
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

3. Techniques of Differentiation

The Chain Rule

Calculus

3. Techniques of Differentiation

The Chain Rule

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2:21 minutes

Problem 3.19

Textbook Question

Find the derivatives of the functions in Exercises 1–42.
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𝓻 = sin √ 2θ

Verified step by step guidance

Identify the function to differentiate: \( r = \sin(\sqrt{2\theta}) \). This is a composition of functions, involving a sine function and a square root function.

Apply the chain rule for differentiation, which states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Here, \( f(x) = \sin(x) \) and \( g(x) = \sqrt{2\theta} \).

Differentiate the outer function \( f(x) = \sin(x) \) with respect to its argument: \( f'(x) = \cos(x) \).

Differentiate the inner function \( g(x) = \sqrt{2\theta} \) with respect to \( \theta \). First, express \( \sqrt{2\theta} \) as \( (2\theta)^{1/2} \) and use the power rule: \( \frac{d}{d\theta}(2\theta)^{1/2} = \frac{1}{2}(2\theta)^{-1/2} \cdot 2 \).

Combine the results using the chain rule: \( \frac{dr}{d\theta} = \cos(\sqrt{2\theta}) \cdot \frac{1}{2}(2\theta)^{-1/2} \cdot 2 \). Simplify the expression to find the derivative.

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Key Concepts

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

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of a composite function. If a function is composed of two or more functions, the chain rule states that the derivative of the outer function is multiplied by the derivative of the inner function. This is particularly useful when dealing with functions that involve nested expressions, such as trigonometric functions of other functions.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. In calculus, these functions are essential for modeling periodic phenomena and are often involved in differentiation and integration. Understanding their properties and derivatives is crucial for solving problems involving angles and oscillatory behavior.