Topic 1: Definition of a Derivative and Its Geometric Interpretation
1\( 2x \)
2\( 3 \)
3\( 0 \)
4\( \frac{1}{4} \)
5\( -\frac{1}{4} \)
6\( 0 \)
7\( 1 \)
8\( e \)
Topic 2: Basic Differentiation Rules
1\( 15x^2 \)
2\( 4x \)
3\( 4 \)
4\( 12x^3 - 4x \)
5\( 5x^4 - 4 \)
6\( -12x^{-3} + 3 \)
7\( -3x^{-4} - 4x^{-5} \)
8\( \frac{1}{3} x^{-2/3} + \frac{2}{3} x^{-5/3} \)
Topic 3: Product and Quotient Rules
1\( x^2 e^x + 2x e^x \)
2\( \frac{1}{(x + 1)^2} \)
3\( x \cos x + \sin x \)
4\( x^3 e^{-x} - 3x^2 e^{-x} \)
5\( -\frac{2x}{(x^2 + 1)^2} \)
6\( 2x \cos x - x^2 \sin x \)
7\( x^2 e^{x^2} + 2x^3 e^{x^2} \)
8\( \frac{e^x (x^2 + 1) - 2x e^x}{(x^2 + 1)^2} \)
Topic 4: Chain Rule and Implicit Differentiation
1\( 6x (x^2 + 1)^2 \)
2\( 2x \cos(x^2) \)
3\( -\frac{x}{y} \)
4\( 3x^2 e^{x^3} \)
5\( \frac{2x}{x^2 + 1} \)
6\( 3x^2 - 6y + 6x \)
7\( 6 \sin^2(x^2) \cos(x^2) \cdot 2x \)
8\( \frac{\cos(x + y) - 2x}{\cos(x + y)} \)
Topic 5: Higher-Order Derivatives
1\( 6x \)
2\( 4x + 3 \)
3\( 24x \)
4\( -\sin x \)
5\( e^x \)
6\( -x^2 \sin x + 4x \cos x - 2 \sin x \)
Topic 6: Applications of Derivatives (Tangent Lines)
1\( y = 2x - 1 \)
2\( 2 \)
3\( y = -3x - 2 \)
4\( y = 1 \) (horizontal)
5\( y = e (x - 1) + e \)
6\( y = 0 \) (horizontal)
Topic 7: Applications of Derivatives (Motion)
1\( 4 \) (m/s)
2\( 0 \) (m/s²)
3\( 1 \) (m/s)
4\( \cos(\pi/2) = 0 \) (m/s²)
5\( 1, 0 \) (m/s, m/s²)
6\( -e^{-1} \) (m/s²)
Topic 8: Applications of Derivatives (Related Rates)
1\( 12 \) (units²/sec)
2\( 6 \) (units²/sec)
3\( 24\pi \) (cm³/sec)
4\( \pi \) (units²/sec)
5\( -\frac{3}{5} \) (units/sec)
6\( 14\pi \) (units³/sec)
Topic 9: Increasing/Decreasing Functions and Extrema
1\( x \geq 0 \)
2\( -1, 0, 1 \)
3\( x \leq -2 \) or \( x \geq 2 \)
4\( \pm \sqrt{2}/2 \) (local max/min)
5\( (0, 1) \cup (1, \infty) \)
6\( \pi/4, 5\pi/4 \) (max), \( 3\pi/4, 7\pi/4 \) (min)
Topic 10: Concavity, Inflection Points, and the Second Derivative Test
1\( x > 0 \)
2\( 6x \)
3\( \pm 1 \) (inflection points)
4\( x < 0 \)
5\( 0 \) (inflection), concave up \( x > 0 \)
6\( -1 \) (max), \( 1 \) (min)
Topic 11: Curve Sketching Using Derivatives
1Parabola, vertex (0, 0)
2\( -1, 0, 1 \)
3Cubic with local max/min at \( \pm \sqrt{2} \)
W-shaped, local max/min at \( \pm \sqrt{2}/2 \)
5Decaying parabola, inflection at 2
6Wavy, max/min at \( \pi/4, 5\pi/4 \)
Topic 12: L'Hôpital’s Rule and Indeterminate Forms
1\( 1 \)
2\( 2 \)
3\( 1 \)
4\( 2 \)
5\( 0 \)
6\( \frac{1}{6} \)