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Quizzes > Mathematics

Derivative Quiz: Test Your Differentiation Skills

Quick, free quiz for calculus differentiation practice. Instant results.

Editorial: Review CompletedCreated By: Fny NynyUpdated Aug 28, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper illustration of a derivative quiz with graphs rates of change tangent lines and math symbols on sky blue background

This derivative quiz helps you check your understanding of rates of change, tangent slopes, and key rules like power, product, quotient, and chain. Get instant feedback as you work through problems and spot topics to review before a test. For targeted practice, try product rule practice problems, explore trig derivatives quiz, or take a broader calculus ii practice quiz.

What is the derivative of f(x) = 7?
7
1
0
x
The derivative of a constant function is always zero by the constant rule. Since 7 does not depend on x, its rate of change is zero. This is one of the simplest rules in differential calculus. .
What is the derivative of f(x) = x^5?
x^4
x^6
5x^5
5x^4
By the power rule, d/dx[x^n] = n x^(n-1). Here, n = 5 gives 5x^4. The power rule is fundamental for polynomial differentiation. .
What is the derivative of f(x) = 3x + 5?
8
3
3x
5
The derivative of 3x is 3 and the derivative of a constant is zero. Sum and constant multiple rules combine to give 3. This is a direct application of basic differentiation rules. .
What is the derivative of f(x) = x^3 - 2x?
x^2 - 2
3x - 2
3x^2 - 2
3x^2 + 2
Differentiate term by term: d/dx[x^3] = 3x^2 and d/dx[-2x] = -2. Combining yields 3x^2 - 2. This uses both the power rule and constant multiple rule. .
What is the derivative of f(x) = e^x?
x e^x
e^x
e
ln(e^x)
The exponential function e^x is unique because its derivative is itself. This property follows from the limit definition of e. It is a cornerstone of differential calculus. Learn about the exponential function.
What is the derivative of f(x) = sin(x)?
-cos(x)
cos(x)
-sin(x)
sin(x)
The derivative of sin(x) is cos(x) by standard trigonometric differentiation. This result is derived from the limit definition of the derivative and small-angle approximations. It's one of the basic derivatives in calculus. .
What is the derivative of f(x) = cos(x)?
sin(x)
-cos(x)
-sin(x)
cos(x)
The derivative of cos(x) is -sin(x) by standard trigonometric rules. This follows from the limit definition and the unit circle interpretation. It's another basic but essential derivative. .
What is the derivative of f(x) = 2x^2 + 4x + 1?
4x + 4
4x^2 + 4
2x^2 + 4
2x + 4
Apply the power rule term by term: d/dx[2x^2] = 4x, d/dx[4x] = 4, and constants vanish. Summing gives 4x + 4. This is a straightforward polynomial differentiation. .
What is the derivative of f(x) = sin(x)·cos(x)?
cos^2(x) - sin^2(x)
sin^2(x) - cos^2(x)
2 sin(x) cos(x)
-sin(x) cos(x)
Use the product rule: (u·v)' = u'v + uv'. Here u=sin(x), v=cos(x). Compute cos(x)·cos(x) + sin(x)·(-sin(x)) to get cos^2(x) - sin^2(x). This also equals cos(2x). .
What is the derivative of f(x) = x^2·sin(x)?
2x sin(x) + x^2 cos(x)
2x cos(x) + x^2 sin(x)
sin(x) + 2x
x^2 cos(x) - 2x sin(x)
By the product rule with u = x^2 and v = sin(x), u' = 2x and v' = cos(x). So f' = 2x·sin(x) + x^2·cos(x). This combines power rule and trigonometric differentiation. .
What is the derivative of f(x) = (x^3 - 1)/x?
3x^2
(2x^3 + 1)/x^2
2x + 1/x^2
3x^2 - x^-2
You can rewrite f(x) = x^2 - x^{-1} or apply the quotient rule. Differentiating yields 2x + x^{-2}. This mixes power rule and negative exponents. .
What is the derivative of f(x) = e^{3x}?
3 e^{3x}
3x e^{3x}
e^{3x}
x e^{3x}
Use the chain rule: derivative of e^{u} is e^{u}·u'. Here u = 3x so u' = 3. Multiply to get 3e^{3x}. Exponential-chain combinations are common. .
What is the derivative of f(x) = ln(2x + 1)?
1/(2x + 1)
(2x + 1)/2
1/(x + 1)
2/(2x + 1)
By the chain rule for ln(u), d/dx[ln(u)] = u'/u. Here u = 2x + 1, so u' = 2. This gives 2/(2x + 1). .
What is the derivative of f(x) = tan(x)?
sec(x) tan(x)
cos^2(x)
tan(x) sec(x)
sec^2(x)
The derivative of tan(x) is sec^2(x), which follows from quotient rule on sin(x)/cos(x). It's a fundamental derivative in trigonometry. .
What is the derivative of f(x) = arctan(x)?
1/(1 - x^2)
1/(1 + x^2)
1/x^2
x/(1 + x^2)
The derivative of arctan(x) is 1/(1 + x^2). This comes from implicit differentiation of y = arctan(x) to xy'. It's a key result for inverse trig functions. .
What is the derivative of f(x) = sin(x^2)?
2x sin(x^2)
2x cos(x^2)
2 cos(x)
cos(x^2)
By the chain rule, outer function is sin(u) and inner is u = x^2 with u' = 2x. Multiply cos(x^2) by 2x. This is a classic chain-rule example. .
What is the second derivative of f(x) = x^3 - 3x^2 + x?
6x + 1
6x - 6
3x^2 - 6x + 1
6x^2 - 6
First derivative: f'(x) = 3x^2 - 6x + 1. Then differentiate again: f''(x) = 6x - 6. This illustrates successive application of the power rule. .
Using implicit differentiation, what is dy/dx if x^2 + y^2 = 1?
x/y
-x/y
y/x
-y/x
Differentiate both sides: 2x + 2y·(dy/dx) = 0, then solve for dy/dx: dy/dx = -x/y. This is a standard implicit differentiation result for a circle. .
For the parametric curve x = t^2, y = t^3, what is dy/dx in terms of t?
3t/2
2t/3
3t^2/2
2t^3/3t
dy/dt = 3t^2 and dx/dt = 2t, so dy/dx = (3t^2)/(2t) = 3t/2. Parametric differentiation uses dy/dx = (dy/dt)/(dx/dt). .
What is the derivative of f(x) = x^x?
x^{x-1} + x^x
x^x ln(x)
x^{x+1}
x^x (ln(x) + 1)
Use logarithmic differentiation: y = x^x implies ln y = x ln x. Differentiate: y'/y = ln x + 1, so y' = x^x(ln x + 1). This handles variable exponent. .
What is the derivative of f(x) = e^{x^2}?
2 e^{x^2}
x e^{x^2}
e^{x^2}
2x e^{x^2}
Apply the chain rule to e^{u} with u = x^2. The derivative is e^{x^2}·2x = 2x e^{x^2}. This is a common exponential-chain example. .
What is the derivative of f(x) = arctan(x^2)?
x/(1 + x^2)
2x/(1 + x^4)
1/(2x + x^2)
2/(1 + x^2)
By the chain rule, outer derivative is 1/(1 + u^2) with u = x^2, and u' = 2x. Multiply to get 2x/(1 + x^4). This combines inverse trig and chain rule. .
A conical tank of height 4 m and base radius 2 m is draining at 3 m³/min. How fast is the water height h changing when h = 1 m?
-12/? m/min
-3?/4 m/min
-?/12 m/min
-9/4 m/min
Volume V = (1/3)?(r^2)h with r/h = 2/4 = 1/2 gives V = ?h^3/12. Differentiate: dV/dt = (?/4)h^2 dh/dt. Solve dh/dt = (dV/dt)/(?h^2/4) = -3·4/(?·1^2) = -12/?. .
What is the derivative of y = x^{x^2}?
x^{x^2}(2 ln(x) + 1)
x^{x^2}(ln(x) + 2x)
x^{2x}(2x ln(x) + x)
x^{x^2}(2x ln(x) + x)
Take ln: ln y = x^2 ln x, differentiate: y'/y = 2x ln x + x. Thus y' = x^{x^2}(2x ln x + x). This uses logarithmic differentiation. .
What is y''(0) if y = sin(x^2)?
-2
2
0
4
y' = 2x cos(x^2); differentiate: y'' = 2 cos(x^2) - 4x^2 sin(x^2). At x=0, cos(0)=1 and the second term vanishes, so y''(0)=2. .
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Study Outcomes

  1. Apply Differentiation Rules -

    Use fundamental differentiation techniques to compute derivatives of polynomial, trigonometric, exponential, and logarithmic functions accurately.

  2. Interpret Instantaneous Rates of Change -

    Translate derivative values into real-world rate-of-change interpretations and understand their significance in various contexts.

  3. Calculate Tangent Line Slopes -

    Determine the slope of tangent lines at specific points by evaluating derivatives and applying the point-slope form.

  4. Analyze Derivative Quiz Problems -

    Develop effective strategies for tackling diverse calculus derivative quiz questions with confidence and precision.

  5. Recognize Common Differentiation Pitfalls -

    Identify and avoid typical mistakes when applying the product, quotient, and chain rules during derivative calculations.

  6. Evaluate Mastery Through Instant Feedback -

    Use immediate quiz results to pinpoint strengths and areas for improvement, guiding your next steps in derivative practice.

Cheat Sheet

  1. Definition of Derivative -

    The derivative f′(x) is defined as the limit limₕ→0 [f(x+h) - f(x)]/h, capturing the instantaneous rate of change at x (MIT OpenCourseWare). For example, f(x)=x² gives f′(x)=limₕ→0 [(x+h)² - x²]/h=2x. Mastering this limit definition is key to acing any derivative quiz.

  2. Power Rule -

    The power rule states d/dx[x❿]=n·x❿❻¹ (Stewart Calculus), so for f(x)=x❵ you get f′(x)=5x❴. A handy mnemonic is "bring down the exponent and subtract one." Expect this rule to pop up repeatedly in your calculus derivative quiz.

  3. Product and Quotient Rules -

    For two functions u(x) and v(x), the product rule is u′v+uv′, and the quotient rule is (v·u′ - u·v′)/v² (Khan Academy). For example, d/dx[x·eˣ]=1·eˣ+x·eˣ= eˣ(1+x). These formulas are essential for tackling more complex derivatives.

  4. Chain Rule -

    The chain rule for composite functions f(g(x)) is f′(g(x))·g′(x) (Coursera Calculus). If h(x)=sin(3x²), then h′(x)=cos(3x²)·6x. Remember "outer derivative times inner derivative" to breeze through an introduction to derivatives quiz.

  5. Tangent Line Equation -

    The slope of the tangent line at x=a is f′(a), and its equation is y - f(a)=f′(a)(x - a) (University of California). For f(x)=x³ at a=2, slope=3·2²=12, so y - 8=12(x - 2). This formula is your best friend in any tangent line slope quiz or instantaneous rate of change quiz.

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