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 3.6.78

Textbook Question

Find dy/dt when x = 1 if y = x² + 7x − 5 and dx/dt = ¹/₃.

Verified step by step guidance

First, identify the given function y = x² + 7x − 5. This is the function for which we need to find the derivative with respect to time, t.

To find dy/dt, use the chain rule. The chain rule states that dy/dt = (dy/dx) * (dx/dt). We need to find dy/dx first.

Differentiate y = x² + 7x − 5 with respect to x to find dy/dx. The derivative of x² is 2x, and the derivative of 7x is 7. Therefore, dy/dx = 2x + 7.

Substitute x = 1 into the expression for dy/dx to find the specific value of the derivative at x = 1. This gives dy/dx = 2(1) + 7.

Now, substitute dy/dx and dx/dt = ¹/₃ into the chain rule formula: dy/dt = (dy/dx) * (dx/dt). Calculate dy/dt using the values obtained in the previous steps.

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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 context, it helps us differentiate y with respect to t by considering the relationship between x and y, and using the chain rule to account for dx/dt.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, it allows us to find dy/dt by multiplying dy/dx by dx/dt.

Substitution

Substitution involves replacing variables with known values to simplify expressions or solve equations. In this problem, after finding the expression for dy/dt, we substitute x = 1 and dx/dt = 1/3 to calculate the specific value of dy/dt at that point, providing a concrete solution to the problem.

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Find the derivatives with respect to x of the following combinations at the given value of x.

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Textbook Question

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.

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Find the derivatives with respect to x of the following combinations at the given value of x.

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" style="max-width: 100%; white-space-collapse: preserve;" width="250">

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Textbook Question

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Roots (Zeros)

a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.

iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

Textbook Question

b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval −2 < x < 2? Give reasons for your answer.

Textbook Question

Temperatures in Fairbanks, Alaska The graph in the accompanying figure shows the average Fahrenheit temperature in Fairbanks, Alaska, during a typical 365-day year. The equation that approximates the temperature on day x is

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

a. On what day is the temperature increasing the fastest?

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Textbook Question

y = 37 sin[(2π/365)(x − 101)] + 25

and is graphed in the accompanying figure.

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