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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1:46 minutes

Problem 3.5.65.b

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counter example.
b. d²/dx² (sin x) = sin x.

Verified step by step guidance

To determine if the statement \( \frac{d^2}{dx^2} (\sin x) = \sin x \) is true, we need to find the second derivative of \( \sin x \).

First, find the first derivative of \( \sin x \). The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).

Next, find the second derivative by differentiating \( \cos x \). The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).

Thus, the second derivative of \( \sin x \) is \( -\sin x \), not \( \sin x \).

Therefore, the statement \( \frac{d^2}{dx^2} (\sin x) = \sin x \) is false. The correct second derivative is \( -\sin x \).

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Key Concepts

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

Second Derivative

The second derivative of a function measures the rate of change of the first derivative. It provides information about the concavity of the function and can indicate points of inflection. In this context, calculating the second derivative of sin x involves differentiating the function twice with respect to x.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. The sine function, sin x, oscillates between -1 and 1 and has a specific pattern of derivatives: the first derivative is cos x, and the second derivative is -sin x.

True/False Statements in Calculus

In calculus, determining the truth of a statement often involves verifying mathematical identities or properties. For the statement d²/dx² (sin x) = sin x, one must compute the second derivative and compare it to sin x to establish its validity, which in this case reveals that the statement is false.