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

Differentials

Calculus

4. Applications of Derivatives

Differentials

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2:39 minutes

Problem 33a

Textbook Question

Finding Functions from Derivatives

In Exercises 31–36, find all possible functions with the given derivative.

a. y′ = −1 / x²

Verified step by step guidance

Step 1: Recognize that the problem asks for the original function given its derivative, y′ = −1 / x². This means we need to find the antiderivative or integral of the given derivative.

Step 2: Set up the integral to find the function y. The integral of y′ with respect to x is ∫(-1/x²) dx.

Step 3: Simplify the integrand. Note that -1/x² can be rewritten as -x⁻², which is a form that is easier to integrate.

Step 4: Apply the power rule for integration. The power rule states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. For -x⁻², n = -2, so integrate using this rule.

Step 5: After applying the power rule, simplify the expression to find the general form of the function y(x). Don't forget to include the constant of integration, C, which represents all possible functions that have the given derivative.

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Key Concepts

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

Antiderivatives

Antiderivatives, or indefinite integrals, are functions that reverse the process of differentiation. To find a function from its derivative, you need to determine its antiderivative. This involves integrating the derivative function, which in this case is y′ = −1/x², to find the original function y(x).

Integration Techniques

Integration techniques are methods used to find antiderivatives. For the derivative y′ = −1/x², you can use the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration. Applying this rule helps find the function whose derivative is given.

Constant of Integration

The constant of integration, denoted as C, is an arbitrary constant added to the antiderivative. It accounts for the fact that differentiation of a constant yields zero, meaning multiple functions can have the same derivative. When finding functions from derivatives, including C ensures all possible functions are considered.