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

Related Rates

Calculus

4. Applications of Derivatives

Related Rates

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2:01 minutes

Problem 3.8.3

Textbook Question

Assume that y = 5x and dx/dt = 2. Find dy/dt

Verified step by step guidance

Start by identifying the given information: y = 5x and dx/dt = 2. We need to find dy/dt.

Recognize that y is a function of x, and both y and x are functions of time t. This implies we need to use the chain rule for differentiation.

Apply the chain rule: The derivative of y with respect to t, dy/dt, can be found using dy/dt = (dy/dx) * (dx/dt).

Differentiate y = 5x with respect to x to find dy/dx. Since y = 5x, dy/dx = 5.

Substitute dy/dx = 5 and dx/dt = 2 into the chain rule formula: dy/dt = 5 * 2.

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Key Concepts

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

Derivative

A derivative represents the rate at which a function is changing at any given point and is a fundamental tool in calculus. In this context, dy/dt and dx/dt are derivatives that describe how y and x change with respect to time t. Understanding derivatives is crucial for solving problems involving rates of change.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It is essential here because y is a function of x, which is itself a function of t. The chain rule allows us to find dy/dt by multiplying the derivative of y with respect to x (dy/dx) by the derivative of x with respect to t (dx/dt).

Linear Functions

A linear function is a function of the form y = mx + b, where m and b are constants. In this problem, y = 5x is a linear function with a slope of 5. Understanding linear functions helps in recognizing that the derivative dy/dx is constant, which simplifies the application of the chain rule to find dy/dt.

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Related Practice

Textbook Question

Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

a. Find the average cost per machine of producing the first 100 washing machines.

Textbook Question

Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Textbook Question

Economics

Marginal revenue

Suppose that the revenue from selling x washing machines is

r(x) = 20000(1 − 1/x) dollars.

b. Use the function r'(x) to estimate the increase in revenue that will result from increasing production from 100 machines a week to 101 machines a week.

Textbook Question

Economics

Marginal revenue

Suppose that the revenue from selling x washing machines is

r(x) = 20000(1 − 1/x) dollars.

c. Find the limit of r'(x) as x → ∞. How would you interpret this number?

Textbook Question

If y = x² and dx/dt = 3, then what is dy/dt when x = –1?

Textbook Question

If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?

Textbook Question

If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?