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

Higher Order Derivatives

Calculus

3. Techniques of Differentiation

Higher Order Derivatives

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2:15 minutes

Problem 3.3.9

Textbook Question

Derivative Calculations

In Exercises 1–12, find the first and second derivatives.

y = 6x² − 10x − 5x⁻²

Verified step by step guidance

Step 1: Identify the function y = 6x² − 10x − 5x⁻². We need to find the first derivative, which involves differentiating each term separately.

Step 2: Differentiate the first term 6x². Using the power rule, the derivative of x^n is n*x^(n-1). Therefore, the derivative of 6x² is 12x.

Step 3: Differentiate the second term -10x. The derivative of x is 1, so the derivative of -10x is -10.

Step 4: Differentiate the third term -5x⁻². Again, using the power rule, the derivative of x⁻² is -2*x⁻³. Therefore, the derivative of -5x⁻² is 10x⁻³.

Step 5: Combine the derivatives from steps 2, 3, and 4 to find the first derivative: y' = 12x - 10 + 10x⁻³. To find the second derivative, differentiate y' using the same rules applied to each term.

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Key Concepts

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

Derivative

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The first derivative indicates how the function is changing, while the second derivative provides insight into the curvature or concavity of the function.

Power Rule

The Power Rule is a basic differentiation technique used to find the derivative of functions in the form of ax^n, where a is a constant and n is a real number. According to this rule, the derivative is calculated by multiplying the coefficient by the exponent and then reducing the exponent by one. This rule simplifies the process of finding derivatives for polynomial functions, making it essential for solving problems involving such expressions.

Concavity and Inflection Points

Concavity refers to the direction in which a function curves, determined by the sign of the second derivative. If the second derivative is positive, the function is concave up, and if it is negative, the function is concave down. Inflection points occur where the concavity changes, which can be identified by setting the second derivative equal to zero. Understanding concavity is crucial for analyzing the behavior of functions and their graphs.