AP - Calculus AB/BC:: Differentiation - Composite, Implicit, and Inverse Functions

The Chain Rule for \(\dfrac{d}{dx}\left[f(g(x))\right]\) states:

- **A)** \(f'(g(x)) + g'(x)\)
- **B)** \(f'(x) \cdot g'(x)\)
- **C)** \(f(g'(x))\)
- **D)** \(f'(g(x)) \cdot g'(x)\)

Use the Chain Rule to find \(\dfrac{d}{dx}\left[(3x+1)^5\right]\).

- **A)** \(5(3x+1)^4 \cdot x\)
- **B)** \(15x^4\)
- **C)** \(15(3x+1)^4\)
- **D)** \(5(3x+1)^4\)

Use the Chain Rule to find \(\dfrac{d}{dx}\left[\sin(x^2)\right]\).

- **A)** \(x^2\cos(x^2)\)
- **B)** \(2x\cos(x^2)\)
- **C)** \(2x\cos(x)\)
- **D)** \(\cos(x^2)\)

Use the Chain Rule to find \(\dfrac{d}{dx}\left[e^{3x}\right]\).

- **A)** \(e^{3}\)
- **B)** \(3e^{3x}\)
- **C)** \(3xe^{3x-1}\)
- **D)** \(e^{3x}\)

Use the Chain Rule to find \(\dfrac{d}{dx}\left[\ln(5x)\right]\).

- **A)** \(5\ln x\)
- **B)** \(\dfrac{1}{5x}\)
- **C)** \(\dfrac{1}{x}\)
- **D)** \(\dfrac{5}{x}\)

Use the Chain Rule to find \(\dfrac{d}{dx}\left[\cos(4x)\right]\).

- **A)** \(-4\sin(4x)\)
- **B)** \(4\sin(4x)\)
- **C)** \(-4\cos(4x)\)
- **D)** \(-\sin(4x)\)

Use the Product Rule together with the Chain Rule to find \(\dfrac{d}{dx}\left[x^2 \sin(3x)\right]\).

- **A)** \(2x\cos(3x)\)
- **B)** \(2x\sin(3x) + 3x^2\cos(3x)\)
- **C)** \(2x\sin(3x) + x^2\cos(3x)\)
- **D)** \(6x\cos(3x)\)

Use the Quotient Rule together with the Chain Rule to find \(\dfrac{d}{dx}\left[\dfrac{\sin(2x)}{x}\right]\).

- **A)** \(\dfrac{2\cos(2x)}{1}\)
- **B)** \(\dfrac{2x\cos(2x) - \sin(2x)}{x^2}\)
- **C)** \(\dfrac{2x\cos(2x) + \sin(2x)}{x^2}\)
- **D)** \(\dfrac{\cos(2x)}{x}\)