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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Multiple Choice

Find the third derivative of the given function.
$y = 3 x^{2} + 9 x + 1$

$0$

$6$

$x$

$3 x squared$

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Verified step by step guidance

Step 1: Begin by identifying the function for which you need to find the third derivative. The function given is $ y = 3x^2 + 9x + 1 $.

Step 2: Calculate the first derivative of the function $ y $ with respect to $ x $. Use the power rule, which states that $ \frac{d}{dx}(x^n) = nx^{n-1} $. Apply this to each term: $ \frac{d}{dx}(3x^2) = 6x $, $ \frac{d}{dx}(9x) = 9 $, and $ \frac{d}{dx}(1) = 0 $. Thus, the first derivative $ y' = 6x + 9 $.

Step 3: Find the second derivative by differentiating the first derivative $ y' = 6x + 9 $. Again, apply the power rule: $ \frac{d}{dx}(6x) = 6 $ and $ \frac{d}{dx}(9) = 0 $. Therefore, the second derivative $ y'' = 6 $.

Step 4: Calculate the third derivative by differentiating the second derivative $ y'' = 6 $. Since $ 6 $ is a constant, its derivative is $ 0 $. Therefore, the third derivative $ y''' = 0 $.

Step 5: Conclude that the third derivative of the function $ y = 3x^2 + 9x + 1 $ is $ 0 $. This indicates that the rate of change of the rate of change of the rate of change of the function is zero, meaning the function is a quadratic polynomial and its third derivative is always zero.