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

4. Applications of Derivatives

Implicit Differentiation

Calculus

4. Applications of Derivatives

Implicit Differentiation

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3:09 minutes

Problem 3.7.16

Textbook Question

Find dr/dθ in Exercises 15–18.

r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴

Verified step by step guidance

Start by identifying the given equation: r - 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴. We need to find dr/dθ, which is the derivative of r with respect to θ.

Rearrange the equation to express r explicitly in terms of θ: r = 2√θ + (3/2)θ²/³ + (4/3)θ³/⁴.

Differentiate each term of the equation with respect to θ. For the term 2√θ, use the derivative of θ^(1/2), which is (1/2)θ^(-1/2).

For the term (3/2)θ²/³, apply the power rule: the derivative is (3/2) * (2/3)θ^(2/3 - 1). Simplify this expression.

For the term (4/3)θ³/⁴, again use the power rule: the derivative is (4/3) * (3/4)θ^(3/4 - 1). Simplify this expression. Combine all the derivatives to find dr/dθ.

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Key Concepts

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

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, r is given implicitly in terms of θ, so we differentiate both sides of the equation with respect to θ, treating r as a function of θ, and then solve for dr/dθ.

Power Rule for Differentiation

The power rule is a basic rule in calculus used to differentiate functions of the form x^n, where n is a real number. The derivative of x^n is n*x^(n-1). In this problem, terms like θ^(2/3) and θ^(3/4) require applying the power rule to find their derivatives with respect to θ.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this context, when differentiating terms involving θ raised to a power, the chain rule helps in handling the differentiation of nested functions, especially when θ is under a radical or fractional exponent.