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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2:33 minutes

Problem 3.7

Textbook Question

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

𝔂 = (θ² + sec θ + 1)³

Verified step by step guidance

Identify the function: We have 𝔂 = (θ² + sec θ + 1)³. This is a composite function, where the outer function is a cube and the inner function is θ² + sec θ + 1.

Apply the chain rule: The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = u³ and g(θ) = θ² + sec θ + 1.

Differentiate the outer function: The derivative of f(u) = u³ with respect to u is f'(u) = 3u². Substitute u = θ² + sec θ + 1 into this derivative.

Differentiate the inner function: Find the derivative of g(θ) = θ² + sec θ + 1 with respect to θ. The derivative of θ² is 2θ, and the derivative of sec θ is sec θ tan θ.

Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function according to the chain rule. This gives you the derivative of the original function 𝔂 with respect to θ.

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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 a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as in the given problem.

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding the secant function is essential when differentiating functions that include it, as it requires the application of the quotient rule and knowledge of trigonometric derivatives. Its behavior and properties are crucial for accurately finding derivatives involving trigonometric functions.

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Related Practice

Textbook Question

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x

Textbook Question

Applying the Chain Rule Use the data in Tables 3.4 and 3.5 of Example 4 to estimate the rate of change in pressure with respect to time experienced by the runner when she is at an altitude of 13,330 ft. Make use of a forward difference quotient when estimating the required derivatives.

Textbook Question

Find the value of dy/dt at t = 0 if y = 3 sin 2x and x = t² + π.

Textbook Question

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

𝔂 = 3 .

(5x² + sin 2x)³/²

Textbook Question

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

𝔂 = 2 tan² x - sec² x

Textbook Question

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

s = cos⁴ (1 - 2t)