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

8. Definite Integrals

Substitution

Calculus

8. Definite Integrals

Substitution

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1:40 minutes

Problem 5.5.1

Textbook Question

On which derivative rule is the Substitution Rule based?

Verified step by step guidance

The Substitution Rule in calculus is based on the Chain Rule for derivatives. Let's explore this connection step by step.

Step 1: Recall the Chain Rule. The Chain Rule states that if a function y = f(g(x)) is composed of two functions, its derivative is given by:

\frac{d}{dx} f (g (x)) = \frac{df}{dg} \cdot \frac{dg}{dx}

Step 2: The Substitution Rule is used in integration to simplify the process of finding antiderivatives. It involves substituting a part of the integrand with a new variable, typically u, where u = g(x).

Step 3: When applying the Substitution Rule, we rewrite the integral in terms of u and du. This process mirrors the Chain Rule because we are essentially reversing the differentiation process.

Step 4: The connection lies in the fact that the derivative of the inner function g(x) (from the Chain Rule) corresponds to the differential du = g'(x) dx in the Substitution Rule.

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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 derivative rule in calculus that allows us to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for the Substitution Rule, as it enables the transformation of variables in integrals and derivatives.

Substitution Rule

The Substitution Rule is a technique used in calculus to simplify the process of finding integrals. It involves substituting a part of the integrand with a new variable, which transforms the integral into a more manageable form. This rule is directly related to the Chain Rule, as it relies on the ability to differentiate composite functions, allowing for easier integration of complex expressions.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In calculus, understanding composite functions is crucial for applying the Chain Rule and the Substitution Rule effectively. Recognizing how to break down complex functions into simpler components allows for easier differentiation and integration, which is the essence of the Substitution Rule.