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Quizzes > Mathematics > Advanced Math

Ultimate Derivative Quiz: Test Your Calculus Skills

Think you can master rates of change? Dive into our calculus derivative quiz!

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 practice core calculus skills: rates of change, tangent slopes, and key rules like power, product, and chain. Use it to spot gaps before an exam; if you want a refresher, skim applications or try extra practice .

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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