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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2:01 minutes

Problem 3.5.34a

Textbook Question

Find y⁽⁴⁾ = d⁴y/dx⁴ if:
a. y = −2 sin x

Verified step by step guidance

Start by identifying the function y = -2 sin(x). We need to find the fourth derivative of this function with respect to x.

Recall that the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). These derivatives alternate as you continue differentiating.

Calculate the first derivative: y' = d/dx [-2 sin(x)] = -2 cos(x).

Calculate the second derivative: y'' = d/dx [-2 cos(x)] = 2 sin(x).

Calculate the third derivative: y''' = d/dx [2 sin(x)] = 2 cos(x). Finally, calculate the fourth derivative: y⁽⁴⁾ = d/dx [2 cos(x)] = -2 sin(x).

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Key Concepts

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

Higher-Order Derivatives

Higher-order derivatives refer to the derivatives of a function taken multiple times. The first derivative represents the rate of change of the function, the second derivative indicates the curvature, and so on. In this case, finding the fourth derivative means applying the differentiation process four times to the function y = -2 sin x.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and describe relationships between angles and sides of triangles. The sine function, in particular, oscillates between -1 and 1 and has specific derivatives: the derivative of sin x is cos x, and the derivative of cos x is -sin x. Understanding these properties is essential for differentiating functions involving trigonometric terms.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. While not directly applied in this specific question, it is crucial for more complex functions involving trigonometric identities.

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