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

Derivatives of Trig Functions

Calculus

3. Techniques of Differentiation

Derivatives of Trig Functions

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Multiple Choice

Differentiate the function: $f (x) = x cos (x) sin (x)$ . Which of the following is the correct derivative $f^{'} (x)$ ?

cos (x) sin (x) - x [{cos}^{2} (x) - {sin}^{2} (x)]

sin (x) cos (x)

cos (x) sin (x) + x [{sin}^{2} (x) + {cos}^{2} (x)]

cos (x) sin (x) + x [{cos}^{2} (x) - {sin}^{2} (x)]

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Verified step by step guidance

Step 1: Recognize that the function f(x) = x * cos(x) * sin(x) is a product of three functions: x, cos(x), and sin(x). To differentiate this, we will use the product rule multiple times.

Step 2: Recall the product rule: If u(x) and v(x) are functions, then the derivative of their product is given by (u * v)' = u' * v + u * v'.

Step 3: Apply the product rule to f(x). First, treat x as one function and cos(x) * sin(x) as another. Differentiate x to get 1, and leave cos(x) * sin(x) unchanged. Then, leave x unchanged and differentiate cos(x) * sin(x).

Step 4: To differentiate cos(x) * sin(x), use the product rule again. Let u(x) = cos(x) and v(x) = sin(x). Differentiate cos(x) to get -sin(x), and differentiate sin(x) to get cos(x). Combine these results: (cos(x) * sin(x))' = -sin^2(x) + cos^2(x).

Step 5: Substitute the results back into the original differentiation process. Combine terms carefully to arrive at the derivative f'(x) = cos(x) * sin(x) + x * [cos^2(x) - sin^2(x)].