Differentiation Rules: math practice, College – download the PDF

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7 October 20267 October 2026

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Math practice College : Differentiation Rules — Zyro the alien explorer of Planète Maths

22 problems graded by difficulty (★ easy, ★★ medium, ★★★ hard). Try first, without looking at the solutions!

Sommaire

1 Product rule on polynomials ★★★

Let \( f(x)=(x^2+1)(x^3-2x) \). Use the product rule to find \( f'(x) \), then check your answer by expanding first.

2 A first quotient ★★★

Let \( g(x)=\dfrac{x}{x+3} \). Find \( g'(x) \) and compute \( g'(1) \).

3 Power of a linear function ★★★

Differentiate \( h(x)=(4x-7)^5 \) and evaluate \( h'(2) \).

4 Trigonometric sum ★★★

Let \( y=3\sin x-2\cos x+\tan x \). Find \( y' \) and evaluate it at \( x=0 \).

5 Exponentials and logarithms ★★★

Differentiate \( f(x)=5e^x-4\ln x+2^x \) for \( x\gt 0 \), and give \( f'(1) \) as an exact value and to two decimal places.

6 Successive derivatives ★★★

For \( f(x)=x^4-3x^3+2x \), find \( f'(x) \), \( f''(x) \), \( f^{(3)}(x) \) and \( f^{(4)}(x) \), then compute \( f''(2) \).

7 Velocity and acceleration of a cart ★★★

A cart on a track has position \( s(t)=t^3-6t^2+9t \) feet after \( t \) seconds. Find its velocity and acceleration at \( t=4 \), and the times when it is momentarily at rest.

8 Product with a sine ★★★

Let \( f(x)=x^2\sin x \). Find \( f'(x) \) and the exact value of \( f'(\pi) \).

9 Where is the tangent horizontal? ★★★

Let \( y=\dfrac{e^x}{x^2+1} \). Show that \( y'=\dfrac{e^x(x-1)^2}{(x^2+1)^2} \) and find every point where the tangent line is horizontal.

10 Cosine of a quadratic ★★★

Differentiate \( y=\cos(3x^2+1) \). Evaluate \( y' \) at \( x=0 \) and at \( x=1 \) (calculator, radians, two decimals).

11 Logarithm of a quadratic ★★★

Let \( y=\ln(x^2+4x+5) \). Explain why the function is defined for every real \( x \), then find \( y' \) and the value of \( x \) where \( y'=0 \).

12 Tangent line to an exponential product ★★★

Let \( f(x)=x\,e^{-2x} \). (a) Find the equation of the tangent line at \( x=0 \). (b) Find the point where the tangent line is horizontal.

13 Implicit slope of a tilted curve ★★★

The point \( (2,1) \) lies on the curve \( x^2+xy+y^2=7 \). Verify this, then find \( \dfrac{dy}{dx} \) and the tangent line at that point.

14 Derivatives of x ln x ★★★

For \( x\gt 0 \) let \( f(x)=x\ln x \). Find \( f' \), \( f'' \) and \( f^{(3)} \), then compute \( f''(4) \) and \( f^{(3)}(2) \).

15 Find the mistakes ★★★

A student wrote: (i) \( \dfrac{d}{dx}(x^2e^x)=2x\,e^x \); (ii) \( \dfrac{d}{dx}\sin(4x)=\cos(4x) \). Explain each error and give the correct derivatives.

16 A variable exponent ★★★

For \( x\gt 0 \) let \( y=x^{\sin x} \). Use logarithmic differentiation to find \( y' \), then compute \( y'\!\left(\dfrac{\pi}{2}\right) \).

17 A long product and quotient ★★★

Let \( y=\dfrac{(x^2+1)^3(2x-1)^4}{(x+5)^2} \) for \( x\gt \tfrac12 \). Use logarithmic differentiation to write \( \dfrac{y'}{y} \), then compute \( y'(1) \).

18 Three layers ★★★

Let \( y=\sqrt{\sin(x^2)} \). Find \( y' \), and evaluate it at \( x=\sqrt{\pi/6} \).

19 Second derivative of a circle ★★★

For the circle \( x^2+y^2=25 \), we know \( y'=-\dfrac xy \). Differentiate again to show that \( y''=-\dfrac{25}{y^3} \), then evaluate at \( (3,4) \) and say what it tells you about the concavity.

20 Inflating balloon ★★★

A spherical balloon has volume \( V=\tfrac43\pi r^3 \) cubic inches. Its radius grows at \( 0.5 \) inch per second. How fast is the volume growing when \( r=6 \) inches? (One inch is 2.54 cm.)