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

Let $f (z) = \sin (z)$ . What is the $43^{r d}$ derivative of $f (z)$ ?

- \sin (z)

\sin (z)

- \cos (z)

\cos (z)

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Verified step by step guidance

Step 1: Recall the derivatives of the sine function. The first derivative of sin(z) is cos(z), the second derivative is -sin(z), the third derivative is -cos(z), and the fourth derivative returns to sin(z). This establishes a repeating cycle every four derivatives: sin(z), cos(z), -sin(z), -cos(z).

Step 2: To find the 43rd derivative, observe that the derivatives repeat in cycles of 4. Divide 43 by 4 to determine the remainder, as the remainder will indicate the position in the cycle.

Step 3: Perform the division: 43 ÷ 4 = 10 remainder 3. This means the 43rd derivative corresponds to the third position in the cycle.

Step 4: Match the remainder to the cycle: The cycle is {sin(z), cos(z), -sin(z), -cos(z)}. A remainder of 3 corresponds to -cos(z).

Step 5: Conclude that the 43rd derivative of f(z) = sin(z) is -cos(z).