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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2:54 minutes

Problem 3.6.75g

Textbook Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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

g. 1 / g²(x), x = 3

Verified step by step guidance

Step 1: Recall the derivative rule for the reciprocal of a function. If h(x) = 1 / g²(x), then h'(x) can be found using the chain rule and the power rule. Specifically, h'(x) = -2 * g(x)^(-3) * g'(x).

Step 2: Substitute the given value of x = 3 into the formula. From the table, g(x) = -4 and g'(x) = 5 at x = 3.

Step 3: Replace g(x) and g'(x) in the derivative formula. This gives h'(x) = -2 * (-4)^(-3) * 5.

Step 4: Simplify the expression step by step. First, calculate (-4)^(-3), which is the reciprocal of (-4)^3. Then multiply by -2 and g'(x).

Step 5: The simplified expression will give the derivative h'(x) at x = 3. Ensure all calculations are consistent with the rules of exponents and multiplication.

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Key Concepts

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

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule helps differentiate expressions involving powers of functions, such as g²(x), by considering the derivative of the outer function and multiplying it by the derivative of the inner function.

Power Rule

The power rule is a basic differentiation rule that states if f(x) = x^n, then f'(x) = n*x^(n-1). This rule is essential for finding the derivative of functions raised to a power, such as g²(x). In the context of the problem, it helps simplify the differentiation process by providing a straightforward method to handle powers of functions.

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two other functions. It states that if h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/g²(x). In this problem, the quotient rule is applied to find the derivative of 1/g²(x), which involves differentiating a function in the form of a reciprocal, requiring careful application of the rule to ensure accuracy.

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Related Practice

Textbook Question

[Technology Exercise]

Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula

y = 6(1 - t/12)² m.

a. Find the rate dy/dt (m/h) at which the tank is draining at time t.

Textbook Question

[Technology Exercise]

Draining a tank It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth y of fluid in the tank t hours after the valve is opened is given by the formula

y = 6(1 - t/12)² m.

b. When is the fluid level in the tank falling fastest? Slowest? What are the values of dy/dt at these times?

Textbook Question

The number of gallons of water in a tank t minutes after the tank has started to drain is Q(t) = 200(30 - t)². How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

Textbook Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.

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