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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2:06 minutes

Problem 3.3.2

Textbook Question

Derivative Calculations

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

y = x² + x + 8

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivatives. In this case, the function is y = x² + x + 8.

Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that if y = xⁿ, then the derivative dy/dx = n*x^(n-1).

Step 3: Apply the power rule to the first term x². The derivative of x² is 2*x^(2-1) = 2x.

Step 4: Apply the power rule to the second term x. The derivative of x is 1*x^(1-1) = 1.

Step 5: The constant term 8 has a derivative of 0, as the derivative of any constant is zero. Combine the derivatives of each term to find the first derivative: dy/dx = 2x + 1. Then, apply the same process to find the second derivative by differentiating dy/dx = 2x + 1 again.

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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 of 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 process of differentiation for functions involving powers of x.

Second Derivative

The second derivative of a function is the derivative of the first derivative. It provides information about the curvature of the function's graph and can indicate whether the function is concave up or concave down. The second derivative is useful for analyzing the acceleration of a function and identifying points of inflection where the function changes concavity.