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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2:05 minutes

Problem 3.23

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(x) = 5x³

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivative. Here, the function is \( f(x) = 5x^3 \).

Step 2: Recall the power rule for differentiation, which states that if \( f(x) = ax^n \), then \( f'(x) = anx^{n-1} \).

Step 3: Apply the power rule to the function \( f(x) = 5x^3 \). Here, \( a = 5 \) and \( n = 3 \).

Step 4: Differentiate the function using the power rule: \( f'(x) = 5 \times 3x^{3-1} \).

Step 5: Simplify the expression obtained from differentiation: \( f'(x) = 15x^2 \).

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Key Concepts

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

Derivatives

A derivative represents the rate at which a function changes at any given point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative provides the slope of the tangent line to the curve of the function at a specific point.

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) = n*x^(n-1). This rule simplifies the process of differentiation for functions involving powers of x.

Function Notation

Function notation is a way to represent mathematical functions in a clear and concise manner. In this context, f(x) denotes a function of x, allowing us to express the relationship between the input x and the output f(x). Understanding function notation is essential for applying calculus concepts, including differentiation.

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Related Practice

Textbook Question

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = g(x) at x = 3.

Textbook Question

Suppose f(3) = 1 and f′(3) = 4. Let g(x) = x² + f(x) and h(x) = 3f(x).

Find an equation of the line tangent to y = h(x) at x = 3.

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

y = x⁵

Textbook Question

The following limits represent f'(a) for some function f and some real number a.

Find a possible function f and number a.

lim x🠂0 e^x-1 / x

Textbook Question

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂0 e^x-1 / x

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

h(t) = t²/2 + 1

Textbook Question

The following limits represent f'(a) for some function f and some real number a.

b. Evaluate the limit by computing f'(a).

lim x🠂1 x¹⁰⁰-1 / x-1

Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

f(v) = v¹⁰⁰+e^v+10

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Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.f(x) = 5x³

Key Concepts

Derivatives

Power Rule

Function Notation

Watch next

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(x) = 5x³