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

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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2:07 minutes

Problem 3.3.59

Textbook Question

The general polynomial of degree n has the form
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀,
where aₙ ≠ 0. Find P'(x).

Verified step by step guidance

To find the derivative of the polynomial P(x), apply the power rule for differentiation, which states that the derivative of xⁿ is n*xⁿ⁻¹.

Start with the first term aₙxⁿ. The derivative is aₙ * n * xⁿ⁻¹.

Proceed to the next term aₙ₋₁xⁿ⁻¹. The derivative is aₙ₋₁ * (n-1) * xⁿ⁻².

Continue applying the power rule to each term: a₂x² becomes 2*a₂*x, and a₁x becomes a₁.

The constant term a₀ has a derivative of 0, as constants disappear when differentiated. Combine all these derivatives to form P'(x).

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Key Concepts

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

Polynomial Function

A polynomial function is an expression consisting of variables and coefficients, structured in terms of powers of a variable. The general form is P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer, and aₙ ≠ 0. Understanding the structure of polynomials is crucial for applying differentiation rules.

Power Rule for Differentiation

The power rule is a basic principle in calculus used to differentiate functions of the form xⁿ. It states that the derivative of xⁿ is nxⁿ⁻¹. This rule is essential for finding the derivative of each term in a polynomial, allowing us to compute the derivative of the entire polynomial function.

Derivative of a Polynomial

The derivative of a polynomial function is obtained by applying the power rule to each term individually. For P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, the derivative P'(x) is aₙnxⁿ⁻¹ + aₙ₋₁(n-1)xⁿ⁻² + ... + a₁. This process involves reducing the power of each term by one and multiplying by the original power.

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Related Practice

Textbook Question

Finding Derivative Functions and Values

Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.

p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Textbook Question

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

f(x) = x + 9/x, x = −3

Textbook Question

Theory and Examples

In Exercises 51–54,

d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)

y = x³/3

Textbook Question

Theory and Examples

In Exercises 51–54,

b. Graph y = f(x) and y = f'(x) side by side using separate sets of coordinate axes, and answer the following questions.

y = x⁴/4

Textbook Question

Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

Textbook Question

The Reciprocal Rule

b. Show that the Reciprocal Rule and the Derivative Product Rule together imply the Derivative Quotient Rule.

Textbook Question

Slopes and Tangent Lines

In Exercises 13–16, differentiate the functions and find the slope of the tangent line at the given value of the independent variable.

y = (x + 3)/(1 – x), x = −2

Textbook Question

In Exercises 19–22, find the values of the derivatives.

dr/dθ |θ₌₀ if r = 2/√(4 – θ)

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