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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2:33 minutes

Problem 3.1

Textbook Question

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

𝔂 = x⁵ - 0.125x² + 0.25x

Verified step by step guidance

Identify the function for which you need to find the derivative: 𝔂 = x⁵ - 0.125x² + 0.25x.

Apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹. Start with the first term, x⁵. The derivative is 5*x⁴.

Move to the second term, -0.125x². Using the power rule, the derivative is -0.125 * 2 * x¹, which simplifies to -0.25x.

Differentiate the third term, 0.25x. The derivative of x is 1, so the derivative of 0.25x is 0.25.

Combine the derivatives of each term to find the derivative of the entire function: 𝔂' = 5x⁴ - 0.25x + 0.25.

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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 practical terms, the derivative provides the slope of the tangent line to the curve of the function at any given point.

Power Rule

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

Sum Rule

The sum rule in calculus states that the derivative of a sum of functions is equal to the sum of their derivatives. If f(x) = g(x) + h(x), then the derivative f'(x) = g'(x) + h'(x). This rule allows for the differentiation of complex functions by breaking them down into simpler components.

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Derivatives Find and simplify the derivative of the following functions.

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

𝔂 = x³ - 3 (x² + π²)

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

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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?

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