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

2. Intro to Derivatives

Derivatives as Functions

Calculus

2. Intro to Derivatives

Derivatives as Functions

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Multiple Choice

If $f$ is a differentiable function and $y = \sin (f (x^{2}))$ , what is $\frac{d y}{d x}$ when $x = 3$ ?

f^{'} (x) \cos (f (x^{2}))

evaluated at

x = 3

\cos (f (x^{2}))

evaluated at

x = 3

2 x f^{'} (x^{2}) \cos (f (x^{2}))

evaluated at

x = 3

2 x f^{'} (x) \sin (f (x^{2}))

evaluated at

x = 3

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Verified step by step guidance

Step 1: Recognize that the problem involves finding the derivative of y = sin(f(x^2)) with respect to x. This requires the chain rule since the function is composed of multiple layers: the sine function, the function f, and the x^2 term.

Step 2: Apply the chain rule. Start by differentiating the outermost function, sin(f(x^2)), with respect to its argument f(x^2). The derivative of sin(u) is cos(u), so the derivative becomes cos(f(x^2)) multiplied by the derivative of f(x^2).

Step 3: Differentiate f(x^2) with respect to x. Since f is a differentiable function, the derivative of f(x^2) is f'(x^2) multiplied by the derivative of x^2. The derivative of x^2 is 2x, so this part becomes 2x f'(x^2).

Step 4: Combine the results from Steps 2 and 3. The derivative dy/dx is given by cos(f(x^2)) multiplied by 2x f'(x^2). This is the complete expression for dy/dx.

Step 5: Evaluate the derivative at x = 3. Substitute x = 3 into the expression 2x f'(x^2) cos(f(x^2)) to find the value of dy/dx at this specific point. Ensure that f'(x^2) and f(x^2) are calculated correctly using the given function f.