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Quizzes > High School Quizzes > Mathematics

Master Derivative Practice Quiz

Sharpen your calculus skills with interactive problems

Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Colorful paper art promoting Derivatives Demystified, a calculus quiz for high school students.

What is the derivative of f(x) = x^3?
3x
x^3
3x^2
2x^2
Using the power rule, the derivative of x^n is n*x^(n-1). For n = 3, the derivative is 3x^2, which is why '3x^2' is correct.
What is the derivative of f(x) = 5x^2?
25x
5x
10x
2x
By applying the power rule to x^2, the derivative is 2x. Multiplying by the constant 5 gives 10x. Therefore, '10x' is the correct answer.
What is the derivative of f(x) = 7?
0
1
x
7
The derivative of any constant is zero because a constant does not change. This property leads directly to the answer '0'.
Find the derivative of f(x) = x^2 + 3x.
3x + 2
2x - 3
2x + 3
x + 3
Differentiate each term separately: the derivative of x^2 is 2x and the derivative of 3x is 3. Adding these together yields 2x + 3, which is the correct answer.
Determine the derivative of f(x) = 2x.
1
2
2x
0
For a linear function f(x) = mx, the derivative is the constant m. Here, m equals 2, so the derivative is simply 2.
Use the product rule: Find the derivative of f(x) = x * sin(x).
cos(x) + x sin(x)
sin(x) - x cos(x)
sin(x) + x cos(x)
x cos(x)
The product rule states that the derivative of u*v is u'v + uv'. Here, u = x (u' = 1) and v = sin(x) (v' = cos(x)), so the derivative becomes sin(x) + x cos(x).
What is the derivative of f(x) = cos(x)?
sin(x)
cos(x)
-sin(x)
-cos(x)
The differentiation rule for cosine is that its derivative is -sin(x). This is a fundamental trigonometric derivative.
Differentiate f(x) = √(2x - 1) (i.e., f(x) = (2x - 1)^(1/2)).
2/√(2x - 1)
1/√(2x - 1)
√(2x - 1)
1/(2√(2x - 1))
Using the chain rule where the outer function is the square root and the inner function is 2x - 1, the derivative is computed as (1/2)(2x - 1)^(-1/2) multiplied by the derivative of 2x - 1, which is 2. This simplifies to 1/√(2x - 1).
Differentiate f(x) = e^(3x).
e^(3x)
e^(x)
3e^(3x)
3e^(x)
When differentiating an exponential function, the chain rule applies. Since the derivative of 3x is 3, the derivative of e^(3x) is 3e^(3x).
Find the derivative of f(x) = ln(x).
x
ln(x)/x
e^x
1/x
The derivative of the natural logarithm ln(x) is 1/x. This is a core result in differentiation and is directly applied here.
Determine the derivative of f(x) = (x^2 + 1)/x.
1 + 1/x^2
1 - 1/x^2
x - 1/x
2x/x^2
By rewriting the function as x + 1/x, the differentiation becomes straightforward. The derivative of x is 1 and the derivative of 1/x is -1/x^2, so the combined derivative is 1 - 1/x^2.
Differentiate f(x) = tan(x).
tan(x) sec(x)
-sec^2(x)
1 + sin^2(x)
sec^2(x)
The derivative of tan(x) is a well-known result: sec^2(x). This is derived either directly or by applying the quotient rule to sin(x)/cos(x).
Differentiate f(x) = 3x^(-2).
6x^(-3)
-6x^(-2)
-6x^(-3)
-3x^(-2)
Applying the power rule, the derivative of x^n is n*x^(n-1). For n = -2, the derivative is -2x^(-3), and multiplying by the constant 3 gives -6x^(-3).
Find the derivative of f(x) = sin^2(x).
cos^2(x)
sin(x) cos(x)
2 sin(x) cos(x)
-2 sin(x) cos(x)
Using the chain rule, differentiate the outer function (squaring) and then multiply by the derivative of sin(x) which is cos(x). The resulting derivative is 2 sin(x) cos(x).
Determine the derivative of f(x) = x ln(x).
1/x
x + ln(x)
ln(x) - 1
ln(x) + 1
Using the product rule, differentiate x (which gives 1) and ln(x) (which gives 1/x). Adding these two results in ln(x) + 1, the correct derivative of the product.
Differentiate f(x) = ln(sin(x)).
cot(x)
tan(x)
1/sin(x)
-cot(x)
The chain rule is used here: the derivative of ln(u) is 1/u times u'. With u = sin(x), its derivative is cos(x), leading to cos(x)/sin(x), which simplifies to cot(x).
Find the derivative of f(x) = e^(x^2).
xe^(x^2)
2xe^(x^2)
e^(x^2)
2e^(x^2)
Applying the chain rule, the derivative of e^(u) is e^(u) multiplied by the derivative of u. Here, u = x^2, whose derivative is 2x. Thus, the final derivative becomes 2xe^(x^2).
Differentiate f(x) = (x^2 + 1)e^(3x).
e^(3x)(3x^2 + 2x + 3)
e^(3x)(2x + 3)
e^(3x)(2x + 3x^2)
e^(3x)(3x^2 + 3)
The product rule is required; differentiate (x^2 + 1) to get 2x and e^(3x) to get 3e^(3x) using the chain rule. Combining these parts results in e^(3x)(2x + 3x^2 + 3).
Determine the derivative of f(x) = (2x - 3)/(x + 4).
11/(x + 4)^2
11/(x - 4)^2
((2x - 3) - (x + 4))/(x + 4)^2
5/(x + 4)
Using the quotient rule, the derivative is given by [(2*(x + 4)) - (2x - 3)*(1)] divided by (x + 4)^2. Simplification of the numerator leads to 11, resulting in the derivative 11/(x + 4)^2.
Find the derivative of f(x) = (sin(x))^3.
sin^3(x) cos(x)
3 sin^2(x) cos(x)
3 sin(x) cos^2(x)
3 cos^2(x) sin(x)
Using the chain rule, treat sin(x) as the inner function raised to the power of 3. Differentiating gives 3 times sin^2(x) multiplied by the derivative of sin(x), which is cos(x), so the answer is 3 sin^2(x) cos(x).
0
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Study Outcomes

  1. Understand the fundamental concept of the derivative and its geometric interpretation.
  2. Compute derivatives for various functions using established differentiation rules.
  3. Apply differentiation techniques to solve practical calculus problems.
  4. Analyze function behavior by identifying critical points and rates of change.
  5. Evaluate problem-solving strategies to enhance test and exam preparation.

Derivative Practice Cheat Sheet

  1. Master the Power Rule - To differentiate x❿, multiply by the exponent and reduce the power by one: d/dx(x❿) = n·x❿❻¹. For instance, d/dx(x³) = 3x². It feels like a magic trick that makes tackling polynomials instantaneously easier! Differentiation Rules - OpenStax
  2. Sum & Difference Rules - Breaking down complex expressions is a snap since d/dx[f(x) ± g(x)] = f′(x) ± g′(x). You simply differentiate each term separately and keep the plus or minus sign. It's as satisfying as assembling building blocks one piece at a time! Differentiation Rules - OpenStax
  3. Utilize the Product Rule - When two functions u(x) and v(x) are multiplied, use d/dx[u·v] = u′·v + u·v′. This handy formula ensures you don't miss a beat when products pop up. Think of it as juggling two balls - keep both in motion with the right rhythm! Product Rule - Wikipedia
  4. Understand the Quotient Rule - For a fraction u(x)/v(x), apply d/dx[u/v] = [u′·v - u·v′] / v². It may look complex, but it's a straightforward subtraction and division routine. Just remember "low d‑high minus high d‑low, over low squared" to ace any fraction! Differentiation Rules - BYJU'S
  5. Grasp the Chain Rule - Composite functions f(g(x)) use d/dx[f∘g] = f′(g(x))·g′(x). You peel back each layer like an onion, differentiating the outer function then plunging into the inner one. It's your go‑to move for nested formulas! Differentiation Rules - BYJU'S
  6. Differentiate Trigonometric Functions - Memorize that d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = - sin(x). Once you lock in these basics, every other trig derivative is a piece of cake. Who knew circles and waves could be so friendly? Derivative Formulas - GeeksforGeeks
  7. Differentiate Exponential Functions - Exponentials are your allies: d/dx[eˣ] = eˣ. That means the function and its derivative are identical - talk about effortless growth! It's like having an endless energy boost in calculus form. Derivative Formulas - GeeksforGeeks
  8. Differentiate Logarithmic Functions - For natural logs, d/dx[ln(x)] = 1/x. It's the neat inverse of exponentials and saves you every time you see ln popping up. Just remember the reciprocal trick and you're golden! Derivative Formulas - GeeksforGeeks
  9. Apply the Constant Multiple Rule - If you have a constant c times a function, d/dx[c·f(x)] = c·f′(x). The constant tags along happily while you do the real work on f(x). It's the ultimate shortcut for scaling derivatives! Derivative Rules - Cuemath
  10. Practice with Real-World Applications - Use these rules to solve rate‑of‑change problems in physics, biology, and economics. Whether you're finding velocity from position or analyzing growth trends, hands‑on practice cements your skills. Grab some real data and watch calculus come to life! Differentiation Rules - OpenStax
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