Derivatives — Calculus I Practice Worksheet

Grades 11–12 / College Hard difficulty 10 problems

The derivative of a function measures its instantaneous rate of change — the slope of the tangent line at any point. Notation includes $f'(x)$ , $\frac{dy}{dx}$ , and $\frac{d}{dx}[f(x)]$ .

Core rules: Power rule $\frac{d}{dx}[x^n] = nx^{n-1}$ ; constant rule $\frac{d}{dx}[c] = 0$ ; sum rule; product rule $(fg)' = f'g + fg'$ ; quotient rule; and chain rule for compositions.

Derivatives appear everywhere in physics (velocity, acceleration), economics (marginal cost), and optimization. Always simplify before differentiating when possible.

Skills practiced

Power rule differentiation
Product and quotient rules
Chain rule for composite functions
Finding equations of tangent lines

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Practice Worksheet: Derivatives

Instructions: Solve each problem carefully. Show all work clearly. Write your final answer in the space provided or on a separate sheet as directed.

1.Find the derivative of $f(x) = \dfrac{x^2 \sin x}{e^x}$ .
2.A particle moves along a line so that its position at time $t$ (in seconds) is given by $s(t) = t^3 - 6t^2 + 9t + 2$ (in meters). Find the velocity of the particle at $t = 2$ seconds.
3.Find $\dfrac{dy}{dx}$ if $y = \ln\left( \sqrt{\dfrac{1 + \cos x}{1 - \cos x}} \right)$ .
4.The radius of a sphere is increasing at a rate of 2 cm/s. How fast is the volume increasing when the radius is 5 cm? (Volume of a sphere: $V = \frac{4}{3}\pi r^3$ )
5.Find the derivative of $g(x) = \arctan\left( \dfrac{x}{1 + x^2} \right)$ .
6.A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall?
7.Find $f'(x)$ if $f(x) = x^{\sin x}$ for $x > 0$ .
8.A conical tank (with vertex down) has a radius of 3 m at the top and a height of 6 m. Water is being pumped into the tank at a rate of 2 m $^3$ /min. How fast is the water level rising when the water is 4 m deep? (Volume of a cone: $V = \frac{1}{3}\pi r^2 h$ )
9.Find the derivative of $h(x) = \sec^{-1}(e^{2x})$ .
10.A plane flying horizontally at an altitude of 1 mile and a speed of 500 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station.

Answer Key

1.

Using quotient rule: $f'(x) = \frac{(2x\sin x + x^2\cos x)e^x - x^2\sin x \cdot e^x}{e^{2x}} = \frac{2x\sin x + x^2\cos x - x^2\sin x}{e^x}$

Final answer: $f'(x) = \dfrac{2x \sin x + x^2 \cos x - x^2 \sin x}{e^x}$
2.

$v(t) = s'(t) = 3t^2 - 12t + 9$ . At $t=2$ , $v(2) = 3(4) - 12(2) + 9 = 12 - 24 + 9 = -3$ m/s (moving downward).

Final answer: $-3$ m/s
3.

Simplify: $y = \frac{1}{2} \ln\left( \frac{1+\cos x}{1-\cos x} \right) = \frac{1}{2} [\ln(1+\cos x) - \ln(1-\cos x)]$ . Then $y' = \frac{1}{2} \left( \frac{-\sin x}{1+\cos x} - \frac{\sin x}{1-\cos x} \right) = \frac{1}{2} \left( \frac{-\sin x(1-\cos x) - \sin x(1+\cos x)}{1-\cos^2 x} \right) = \frac{1}{2} \left( \frac{-2\sin x}{\sin^2 x} \right) = -\csc x$ .

Final answer: $-\csc x$
4.

$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$ . Given $\frac{dr}{dt} = 2$ cm/s, $r=5$ : $\frac{dV}{dt} = 4\pi (25)(2) = 200\pi$ cm $^3$ /s.

Final answer: $200\pi$ cm $^3$ /s
5.

$g'(x) = \frac{1}{1 + \left( \frac{x}{1+x^2} \right)^2} \cdot \frac{(1)(1+x^2) - x(2x)}{(1+x^2)^2} = \frac{1}{1 + \frac{x^2}{(1+x^2)^2}} \cdot \frac{1+x^2 - 2x^2}{(1+x^2)^2} = \frac{1}{\frac{(1+x^2)^2 + x^2}{(1+x^2)^2}} \cdot \frac{1 - x^2}{(1+x^2)^2} = \frac{(1+x^2)^2}{(1+x^2)^2 + x^2} \cdot \frac{1 - x^2}{(1+x^2)^2} = \frac{1 - x^2}{(1+x^2)^2 + x^2}$ .

Final answer: $g'(x) = \dfrac{1 - x^2}{(1+x^2)^2 + x^2}$
6.

Let $x$ be distance from wall to bottom, $y$ be height of top on wall. $x^2 + y^2 = 100$ . Differentiate: $2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \Rightarrow \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}$ . When $x=6$ , $y=8$ , $\frac{dx}{dt}=1$ : $\frac{dy}{dt} = -\frac{6}{8}(1) = -\frac{3}{4}$ ft/s (sliding down at 0.75 ft/s).

Final answer: $-\frac{3}{4}$ ft/s
7.

$f(x) = e^{\sin x \ln x}$ . Then $f'(x) = e^{\sin x \ln x} \left( \cos x \ln x + \frac{\sin x}{x} \right) = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right)$ .

Final answer: $f'(x) = x^{\sin x} \left( \cos x \ln x + \frac{\sin x}{x} \right)$
8.

By similar triangles, $\frac{r}{h} = \frac{3}{6} = \frac{1}{2}$ , so $r = \frac{h}{2}$ . Then $V = \frac{1}{3}\pi \left( \frac{h}{2} \right)^2 h = \frac{\pi}{12} h^3$ . Differentiate: $\frac{dV}{dt} = \frac{\pi}{4} h^2 \frac{dh}{dt}$ . Given $\frac{dV}{dt}=2$ , $h=4$ : $2 = \frac{\pi}{4}(16) \frac{dh}{dt} \Rightarrow \frac{dh}{dt} = \frac{2}{4\pi} = \frac{1}{2\pi}$ m/min.

Final answer: $\frac{1}{2\pi}$ m/min
9.

$h'(x) = \frac{1}{|e^{2x}| \sqrt{(e^{2x})^2 - 1}} \cdot 2e^{2x} = \frac{2e^{2x}}{e^{2x} \sqrt{e^{4x} - 1}} = \frac{2}{\sqrt{e^{4x} - 1}}$ .

Final answer: $h'(x) = \dfrac{2}{\sqrt{e^{4x} - 1}}$
10.

Let $x$ be horizontal distance from station to plane, $s$ be distance from station to plane. $s^2 = x^2 + 1^2$ . Differentiate: $2s \frac{ds}{dt} = 2x \frac{dx}{dt} \Rightarrow \frac{ds}{dt} = \frac{x}{s} \frac{dx}{dt}$ . When $s=2$ , $x = \sqrt{4-1} = \sqrt{3}$ , $\frac{dx}{dt}=500$ : $\frac{ds}{dt} = \frac{\sqrt{3}}{2}(500) = 250\sqrt{3}$ mi/h.

Final answer: $250\sqrt{3}$ mi/h

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Common mistakes to avoid

Forgetting the chain rule on nested functions
Product rule errors (differentiating only one factor)
Leaving negative exponents unconverted before applying power rule
Confusing average rate of change with instantaneous derivative

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Last updated: 2026

Derivatives — Calculus I Practice Worksheet

Skills practiced

Answer Key

Common mistakes to avoid

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