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

Product and Quotient Rules

Calculus

3. Techniques of Differentiation

Product and Quotient Rules

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2:39 minutes

Problem 3.4.15

Textbook Question

Given that f(1) = 5, f′(1) = 4, g(1) = 2, and g′(1) = 3 , find d/dx (f(x)g(x))∣ ∣x=1 and d/dx (f(x) / g(x)) ∣ x=1.

Verified step by step guidance

Step 1: To find the derivative of the product of two functions, use the product rule. The product rule states that if you have two functions f(x) and g(x), then the derivative of their product is given by: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).

Step 2: Apply the product rule to the given functions at x = 1. Substitute f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3 into the product rule formula: (f(x)g(x))'|_{x=1} = f'(1)g(1) + f(1)g'(1).

Step 3: To find the derivative of the quotient of two functions, use the quotient rule. The quotient rule states that if you have two functions f(x) and g(x), then the derivative of their quotient is given by: (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.

Step 4: Apply the quotient rule to the given functions at x = 1. Substitute f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3 into the quotient rule formula: (f(x)/g(x))'|_{x=1} = (f'(1)g(1) - f(1)g'(1)) / (g(1))^2.

Step 5: Simplify the expressions obtained from the product and quotient rules to find the derivatives at x = 1.

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Key Concepts

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

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for solving the first part of the question, where we need to find the derivative of the product f(x)g(x) at x=1.

Quotient Rule

The Quotient Rule is another important differentiation rule that applies when dividing two functions. It states that if you have a function h(x) = f(x)/g(x), the derivative is given by (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. This rule is crucial for solving the second part of the question, where we need to find the derivative of the quotient f(x)/g(x) at x=1.

Evaluation of Derivatives

Evaluating derivatives at a specific point involves substituting the value of x into the derivative expression after applying the appropriate differentiation rules. In this question, we will substitute x=1 into the results obtained from the Product and Quotient Rules to find the specific values of the derivatives at that point, which is necessary for completing the problem.