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

Previous problemNext problem

2:42 minutes

Problem 3.3.6

Textbook Question

Derivative Calculations

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

y = x³/3 + x²/2 + x/4

Verified step by step guidance

Step 1: Identify the function for which you need to find the derivatives. The given function is \( y = \frac{x^3}{3} + \frac{x^2}{2} + \frac{x}{4} \).

Step 2: To find the first derivative, apply the power rule to each term of the function. The power rule states that the derivative of \( x^n \) is \( nx^{n-1} \).

Step 3: Differentiate each term separately: \( \frac{d}{dx} \left( \frac{x^3}{3} \right) = x^2 \), \( \frac{d}{dx} \left( \frac{x^2}{2} \right) = x \), and \( \frac{d}{dx} \left( \frac{x}{4} \right) = \frac{1}{4} \). Combine these results to get the first derivative.

Step 4: To find the second derivative, differentiate the first derivative. Apply the power rule again to each term of the first derivative.

Step 5: Differentiate each term of the first derivative: \( \frac{d}{dx} (x^2) = 2x \), \( \frac{d}{dx} (x) = 1 \), and \( \frac{d}{dx} \left( \frac{1}{4} \right) = 0 \). Combine these results to get the second derivative.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

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

Power Rule for Derivatives

The power rule is a basic principle in calculus used to find the derivative of a function of the form f(x) = x^n. The rule states that the derivative, f'(x), is n*x^(n-1). This rule simplifies the process of differentiation, allowing us to easily find the rate of change of polynomial functions.

Sum Rule for Derivatives

The sum rule for derivatives states that the derivative of a sum of functions is the sum of their derivatives. If you have a function y = f(x) + g(x), the derivative y' is f'(x) + g'(x). This rule allows us to differentiate each term in a polynomial separately and then combine the results.

Second Derivative

The second derivative of a function is the derivative of the first derivative, providing information about the curvature or concavity of the original function. It is denoted as f''(x) or d²y/dx². Calculating the second derivative helps in understanding the acceleration or the rate of change of the rate of change of the function.