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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2:34 minutes

Problem 3.3

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x³ - 3 (x² + π²)

Verified step by step guidance

Identify the function to differentiate: 𝔂 = x³ - 3(x² + π²).

Apply the power rule to differentiate x³. The power rule states that the derivative of x^n is n*x^(n-1).

Differentiate the term -3(x² + π²). Use the constant multiple rule, which allows you to take the constant outside the derivative, and then apply the power rule to x².

Recognize that π² is a constant, and the derivative of a constant is zero.

Combine the derivatives of each term to find the derivative of the entire function.

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Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) or dy/dx, and it provides critical information about the function's behavior, such as its slope and points of tangency.

Power Rule

The power rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) is given by n*x^(n-1). This rule simplifies the differentiation process, allowing for quick calculations of derivatives for terms involving powers of x.

Constant Rule

The constant rule in differentiation states that the derivative of a constant is zero. This means that if a function includes a constant term, it does not affect the slope of the function at any point. For example, in the function y = x³ - 3(x² + π²), the term -3π² is a constant, and its derivative contributes nothing to the overall derivative of the function.

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Textbook Question

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Textbook Question

Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

Textbook Question

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x⁵ - 0.125x² + 0.25x

Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

a. 6ƒ(x) - g(x), x = 1

Textbook Question

Slopes and Tangent Lines

a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

Textbook Question

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Textbook Question

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

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