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

Basic Rules of Differentiation

Calculus

3. Techniques of Differentiation

Basic Rules of Differentiation

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1:23 minutes

Problem 3.3.7

Textbook Question

Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).

Verified step by step guidance

Step 1: Understand the problem. We are given the derivatives of two functions, f and g, at x = 3. Specifically, f'(3) = 6 and g'(3) = -2. We need to find the derivative of the sum of these functions, (f+g)'(3).

Step 2: Recall the rule for the derivative of a sum. The derivative of the sum of two functions is the sum of their derivatives. Mathematically, this is expressed as (f+g)'(x) = f'(x) + g'(x).

Step 3: Apply the rule to the given point. Substitute x = 3 into the formula: (f+g)'(3) = f'(3) + g'(3).

Step 4: Substitute the given values. We know f'(3) = 6 and g'(3) = -2, so substitute these values into the equation: (f+g)'(3) = 6 + (-2).

Step 5: Simplify the expression. Combine the values to find the derivative of the sum at x = 3.

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Key Concepts

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

Derivative of a Function

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and represents the slope of the tangent line to the function's graph at that point. Understanding derivatives is crucial for analyzing how functions behave locally.

Sum Rule of Derivatives

The Sum Rule states that the derivative of the sum of two functions is equal to the sum of their derivatives. Mathematically, if f and g are functions, then (f + g)'(x) = f'(x) + g'(x). This rule simplifies the process of finding derivatives when dealing with combined functions.

Evaluating Derivatives at a Point

To evaluate the derivative of a function at a specific point, you substitute the point's value into the derivative function. In this case, knowing f'(3) and g'(3) allows us to find (f + g)'(3) by directly applying the Sum Rule and substituting the given values.