AP (Advanced Placement) · Calculus Ab

Differentiation

82 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

All 82 questions

If $f (x) = 5 x^{3} + 2 x^{2} + 5 x$ , what is $f^{'} (3)$ ?
If $f (x) = 5 x^{3} - 3 x^{2} + 2 x$ , what is $f^{'} (3)$ ?
If $f (x) = 2 x^{3} + 1 x^{2} + 4 x$ , what is $f^{'} (- 2)$ ?
If $f (x) = 3 x^{3} + 3 x^{2} + 4 x$ , what is $f^{'} (2)$ ?
If $f (x) = 5 x^{3} + 5 x^{2} - 2 x$ , what is $f^{'} (- 2)$ ?
If $f (x) = 5 x^{3} - 2 x^{2} + 3 x$ , what is $f^{'} (- 2)$ ?
If $f (x) = 1 x^{3} + 4 x^{2} + 2 x$ , what is $f^{'} (4)$ ?
If $f (x) = 2 x^{3} + 6 x^{2} + 3 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 5 x^{3} + 5 x^{2}$ , what is $f^{'} (4)$ ?
If $f (x) = 2 x^{3} + 5 x^{2} + 5 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 5 x^{3} + 5 x^{2} - 4 x$ , what is $f^{'} (3)$ ?
If $f (x) = 5 x^{3} + 5 x^{2} + 4 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 3 x^{3} - 2 x^{2} + 4 x$ , what is $f^{'} (1)$ ?
If $f (x) = 4 x^{3} + 3 x^{2} - 5 x$ , what is $f^{'} (2)$ ?
If $f (x) = 2 x^{3} + 6 x^{2} - 2 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 3 x^{3} + 5 x^{2} - 1 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 3 x^{3} + 6 x^{2} - 1 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 5 x^{3} - 4 x^{2} - 2 x$ , what is $f^{'} (- 1)$ ?
If $f (x) = 3 x^{3} - 3 x^{2}$ , what is $f^{'} (0)$ ?
If $f (x) = 5 x^{3} - 2 x^{2} + 4 x$ , what is $f^{'} (3)$ ?
If $f (x) = 3 x^{3} - 5 x^{2} - 4 x$ , what is $f^{'} (0)$ ?
If $f (x) = 3 x^{3} - 2 x$ , what is $f^{'} (4)$ ?
If $f (x) = 2 x^{3} + 6 x^{2} - 1 x$ , what is $f^{'} (0)$ ?
If $f (x) = 4 x^{3} - 2 x^{2} - 3 x$ , what is $f^{'} (3)$ ?
If $f (x) = 3 x^{2} - 1 x$ , find $f^{'} (2)$ .
If $f (x) = 2 x^{4} - 5 x$ , find $f^{'} (2)$ .
If $f (x) = 3 x^{3} - 3 x$ , find $f^{'} (3)$ .
If $f (x) = 3 x^{3} + 3 x$ , find $f^{'} (2)$ .
If $f (x) = 1 x^{3} - 2 x$ , find $f^{'} (3)$ .
If $f (x) = 1 x^{2} + 2 x$ , find $f^{'} (1)$ .
If $f (x) = 4 x^{2} - 1 x$ , find $f^{'} (2)$ .
If $f (x) = 5 x^{3} + 5 x$ , find $f^{'} (3)$ .
If $f (x) = 1 x^{3} + 0 x$ , find $f^{'} (2)$ .
If $f (x) = 1 x^{2} - 1 x$ , find $f^{'} (2)$ .
If $f (x) = 1 x^{4} - 2 x$ , find $f^{'} (3)$ .
If $f (x) = 2 x^{4} + 4 x$ , find $f^{'} (1)$ .
If $f (x) = 5 x^{3} + 5 x$ , find $f^{'} (1)$ .
If $f (x) = 4 x^{4} - 5 x$ , find $f^{'} (3)$ .
If $f (x) = x^{x}$ for $x > 0$ , what is $f^{'} (2)$ ?
A curve is defined by $x^{2} + x y + y^{3} = 11$ . What is $\frac{d y}{d x}$ at the point $(1, 2)$ ?
Air is pumped into a spherical balloon so that its volume increases at a constant $36 π$ cubic centimeters per second. At the instant when the balloon's surface area is $144 π$ square centimeters, a…
What is the absolute maximum value of $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ on the closed interval $[0, 5]$ ?
Let $f (x) = x^{3} + 2 x + 1$ , which is one-to-one. If $g$ is the inverse function of $f$ , what is $g^{'} (4)$ ?
The graph of $y = x e^{- x}$ has a point of inflection. What is its $y$ -coordinate?
Let $F (x) = \int_{x}^{x^{2}} 1 + t^{3} d t$ for $x > 0$ . What is $F^{'} (1)$ ?
If $f (x) = x^{2} e^{- x}$ , what is $f^{''} (1)$ ?
A rectangle has its base on the $x$ -axis and its upper two vertices on the parabola $y = 12 - x^{2}$ , symmetric about the $y$ -axis. What is the maximum possible area of the rectangle?
A curve is given by $x = t^{2} + 1$ and $y = t^{3} - 3 t$ . What is $\frac{d ^{2} y}{d x ^{2}}$ at $t = 2$ ?
Find the derivative of $f (x) = e^{3 x} cos (2 x)$ .
If $y = ln (x^{2} + 4)$ , find $\frac{d y}{d x}$ at $x = 1$ .
Find $\frac{d y}{d x}$ for the curve defined implicitly by $x^{2} + y^{2} = 25$ at the point $(3, 4)$ .
What is the slope of the line tangent to $f (x) = tan (3 x)$ at $x = \frac{π}{12}$ ?
Find the equation of the tangent line to $f (x) = x^{3} - 2 x + 1$ at $x = 1$ .
If $f (x) = \frac{2 x - 1}{x + 3}$ , what is $f^{'} (1)$ ?
Find the critical points of $f (x) = 2 x^{3} - 9 x^{2} + 12 x + 5$ .
On which interval is the function $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ concave downward?
Find the inflection point of the graph of $f (x) = x^{3} - 3 x^{2} + 4$ .
Let $f$ be continuous on $[1, 4]$ and differentiable on $(1, 4)$ . If $f (1) = 2$ and $f (4) = 11$ , what does the Mean Value Theorem guarantee?
A 10-foot ladder leans against a vertical wall. If the bottom of the ladder slides away from the wall at $2$ ft/s, how fast is the top of the ladder sliding down the wall when the bottom is $6$ ft fro…
Find the derivative of $g (x) = arctan (2 x)$ .
If $f (x) = x^{x}$ for $x > 0$ , what is $f^{'} (x)$ ?
Use linear approximation to estimate $9.2$ using the tangent line to $f (x) = x$ at $a = 9$ .
If $f (x) = \int_{0}^{x^{2}} 1 + t^{3} d t$ , find $f^{'} (x)$ .
Find the absolute maximum value of $f (x) = - x^{2} + 4 x + 1$ on the closed interval $[0, 3]$ .
If a particle moves along a line with position function $s (t) = t^{3} - 6 t^{2} + 9 t$ for $t \geq 0$ , at what times is the particle at rest?
Evaluate $lim_{x \to 0} \frac{e ^{2 x} - 1 - 2 x}{x ^{2}}$ using L'Hopital's Rule.
If $f (x) = sin^{2} (3 x)$ , what is $f^{'} (x)$ ?
Let $f (x) = ln (x^{2} + 1)$ . What is the value of $f^{''} (1)$ ?
A 13-foot ladder leans against a vertical wall. If the bottom slides away at 2 feet per second, how fast is the top sliding down when the bottom is 5 feet from the wall?
What is the slope of the line tangent to the curve $x^{3} + y^{3} = 6 x y$ at the point $(3, 3)$ ?
Let $f (x) = x^{4} - 4 x^{3} + 10$ . On which open interval is the graph of $f$ concave downward?
If $f (x) = arcsin (2 x)$ , what is the derivative $f^{'} (x)$ ?
What is $lim_{x \to 0} \frac{sin ( 5 x )}{tan ( 3 x )}$ ?
At what point on the curve $y = x$ is the tangent line parallel to the secant line passing through $(1, 1)$ and $(4, 2)$ ?
Let $f$ be a differentiable function such that $f (2) = 3$ and $f^{'} (2) = - 4$ . What is the linear approximation of $f (2.05)$ ?
If $y = e^{- 3 x} cos (2 x)$ , what is $\frac{d y}{d x}$ at $x = 0$ ?
A particle moves along the $x$ -axis with position $x (t) = t^{3} - 6 t^{2} + 9 t + 1$ . At what time $t > 0$ is the particle momentarily at rest?
What is the absolute minimum value of $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 5$ on the closed interval $[0, 3]$ ?
If $f (x) = \frac{x}{x - 2}$ , what is $f^{'} (3)$ ?
Let $f$ be continuous on $[1, 5]$ and differentiable on $(1, 5)$ with $f (1) = 2$ and $f (5) = 14$ . The Mean Value Theorem guarantees a value $c \in (1, 5)$ where $f^{'} (c)$ equals what?
What is the derivative of $f (x) = x^{2} tan (x)$ at $x = π$ ?
Let $h (x) = f (g (x))$ . If $g (1) = 3$ , $g^{'} (1) = 5$ , and $f^{'} (3) = - 2$ , what is $h^{'} (1)$ ?

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