Derivatives and Differentiation Quiz

Grade 12 · Math · Quiz

Test your mastery of differentiation rules including the Chain Rule, Product Rule, implicit differentiation, and related applications.

Questions

  1. What is the derivative of f(x) = (x² + 1)⁴?

    1. 4(x² + 1)³
    2. 8x(x² + 1)³
    3. 4x(x² + 1)³
    4. 8(x² + 1)³
    Show answer

    Answer: 8x(x² + 1)³

    Chain Rule: outer (u⁴)' = 4u³, inner (x²+1)' = 2x. So 4(x²+1)³ · 2x = 8x(x²+1)³.

  2. Find f'(x) if f(x) = (2x + 3) / (x² - 1).

    1. (2(x² - 1) - 2x(2x + 3)) / (x² - 1)²
    2. (2x² - 2 - 4x² - 6x) / (x² - 1)²
    3. Both A and B are equivalent
    4. 2 / (2x)
    Show answer

    Answer: Both A and B are equivalent

    Quotient Rule gives (2(x²-1) - (2x+3)(2x)) / (x²-1)². Expanding: (2x²-2-4x²-6x)/(x²-1)². Both options A and B simplify to the same expression.

  3. Use implicit differentiation on x² + y² = 25. Find dy/dx.

    1. x/y
    2. -x/y
    3. y/x
    4. -y/x
    Show answer

    Answer: -x/y

    Differentiating implicitly: 2x + 2y(dy/dx) = 0, so dy/dx = -2x/(2y) = -x/y.

  4. If g(x) = e^(x² - 3x), then g'(x) is:

    1. e^(x² - 3x)
    2. (2x - 3)e^(x² - 3x)
    3. 2x · e^(x² - 3x)
    4. (x² - 3x)e^(x² - 3x - 1)
    Show answer

    Answer: (2x - 3)e^(x² - 3x)

    Chain Rule on e^u where u = x²-3x: g'(x) = e^(x²-3x) · (2x-3).

  5. At which x-value does the function f(x) = x³ - 3x² + 2 have a local maximum?

    1. x = 0
    2. x = 2
    3. x = -1
    4. x = 1
    Show answer

    Answer: x = 0

    f'(x) = 3x²-6x = 3x(x-2). Critical points at x=0 and x=2. f''(x)=6x-6; f''(0)=-6 < 0 so x=0 is a local maximum.

  6. The second derivative f''(x) of f(x) = 4x³ - 6x² + x is:

    1. 12x - 6
    2. 24x - 12
    3. 12x² - 12x + 1
    4. 8x - 6
    Show answer

    Answer: 24x - 12

    f'(x) = 12x²-12x+1; f''(x) = 24x-12.

  7. If f''(x) > 0 on an interval, then f is concave up on that interval.

    Show answer

    Answer: True

    The second derivative test for concavity: f''(x) > 0 means the slope of f' is increasing, which corresponds to concave up (cup-shaped).

  8. The derivative of f(x) = cos(x) is ___.

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    Answer: -sin(x)

    The derivative of cosine is negative sine: d/dx[cos x] = -sin x.

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