AP - Calculus AB/BC:: Differentiation - Definition and Fundamental Properties

Which limit expression correctly defines \(f'(a)\), the derivative of \(f\) at \(x=a\)?

- **A)** \(\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}\)
- **B)** \(\lim_{h \to 0} \dfrac{f(a) - f(a+h)}{a}\)
- **C)** \(\lim_{h \to \infty} \dfrac{f(a+h)}{h}\)
- **D)** \(\dfrac{f(a+h) - f(a)}{h}\) for any fixed small \(h\)

The alternate (equivalent) limit definition of \(f'(a)\) is:

- **A)** \(\lim_{x \to 0} \dfrac{f(x)}{x - a}\)
- **B)** \(\lim_{x \to a} f(x) - f(a)\)
- **C)** \(\lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}\)
- **D)** \(\lim_{x \to a} \dfrac{f(a)}{x}\)

Which of the following is NOT standard notation for the derivative of \(y = f(x)\)?

- **A)** \(y'\)
- **B)** \(\dfrac{dy}{dx}\)
- **C)** \(f'(x)\)
- **D)** \(\Delta y\)

The derivative \(f'(a)\) is best interpreted as:

- **A)** The total change in \(f\) from \(0\) to \(a\)
- **B)** The average rate of change of \(f\) over \([a, a+1]\)
- **C)** The instantaneous rate of change of \(f\) at \(x=a\), i.e., the slope of the tangent line to the graph at that point
- **D)** The value of \(f\) at \(x=a\)

Using the Power Rule, find \(\dfrac{d}{dx}\left[x^5\right]\).

- **A)** \(x^4\)
- **B)** \(4x^5\)
- **C)** \(5x^4\)
- **D)** \(5x^5\)

Find \(\dfrac{d}{dx}\left[x^{-3}\right]\).

- **A)** \(-3x^{-2}\)
- **B)** \(-3x^{-4}\)
- **C)** \(3x^{-4}\)
- **D)** \(-\dfrac{1}{3}x^{-4}\)

Find \(\dfrac{d}{dx}\left[\sqrt{x}\right]\) by rewriting \(\sqrt{x} = x^{1/2}\) and applying the Power Rule.

- **A)** \(\dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}\)
- **B)** \(2x^{-1/2}\)
- **C)** \(\dfrac{1}{2}x^{1/2}\)
- **D)** \(\dfrac{1}{2}\)

What is \(\dfrac{d}{dx}[7]\) (the derivative of a constant)?

- **A)** \(0\)
- **B)** \(1\)
- **C)** Undefined
- **D)** \(7\)