Derivative Quiz
Questions: 16 · 10 minutes
1. Suppose f is differentiable and one-to-one, with f(2) = 5 and f′(2) = 4. What is (f⁻¹)′(5)?
4, because the inverse has the same derivative as the original function
1/4, because the inverse derivative is the reciprocal of f′(2)
1/5, because the inverse is evaluated at an output value of 5
2, because f maps the input 2 to the output 5
2. For f(x) = x⁴ − 4x², what does the second derivative reveal about the graph at x = 0?
The graph is concave up there because f″(0) = 8.
The graph is concave down there because f″(0) = −8.
The graph has zero concavity there because f″(0) = 0.
Concavity cannot be assessed there because f′(0) = 0.
3. A point moves along the circle x² + y² = 25. What is the slope dy/dx at the point (3, 4)?
7/2, obtained by differentiating x² + y² as 2x + 2y
3/4, obtained by omitting the negative sign after implicit differentiation
−4/3, obtained by reversing the roles of x and y
−3/4, obtained from dy/dx = −x/y
4. For g(x) = (3x − 1)⁴, a student writes g′(x) = 4(3x − 1)³. What adjustment is required?
Multiply by the inner derivative 3, giving 12(3x − 1)³.
Differentiate the inner expression by subtracting 1, giving 4(3x − 2)³.
Multiply by the original exponent a second time, giving 16(3x − 1)³.
Raise the inner expression to the fourth power again, giving 4(3x − 1)⁴.
5. A student claims that the derivative of x² sin(x) is 2x cos(x). Which expression correctly applies the product rule?
2x sin(x) + x² cos(x), because each factor is differentiated in one product term
2x cos(x), because the derivatives of the two factors are multiplied
x² cos(x), because only the sine factor is differentiated
2x sin(x) − x² cos(x), because differentiating sine changes its sign
6. A function is continuous on [1, 4], differentiable on (1, 4), and has an average rate of change of 6 across [1, 4]. What does the Mean Value Theorem guarantee?
The derivative equals 6 at both endpoints of the closed interval.
The function's output equals 6 at some interior point.
The derivative remains equal to 6 throughout the entire open interval.
There is at least one c in (1, 4) for which f′(c) = 6.
7. A circle's radius increases at 0.5 unit per second. How quickly is its area increasing when the radius is 4 units?
2π square units per second, from omitting the radius in the area-rate formula
4π square units per second, from dA/dt = 2πr · dr/dt
8π square units per second, from treating the radius rate as 1 unit per second
16π square units per second, from reporting the area rather than its rate of change
8. Use the linear approximation of f(x) = √x at x = 4 to estimate √4.1.
2.0125, using a slope of 1/8 at x = 4
2.05, using a slope of 1/2 at x = 4
2.025, using L(x) = 2 + (1/4)(x − 4)
2.1, adding the entire input change directly to √4
9. Let P(x) = x⁵ − 3x² + 7. What is the value of P′(2)?
68, found from P′(x) = 5x⁴ − 6x
61, found by retaining the constant term while differentiating the powers
74, found from P′(x) = 5x⁴ − 3x
128, found by substituting 2 into the original polynomial before differentiating
10. Which values are the critical numbers of f(x) = x³ − 3x?
x = 0 only, because the cubic term vanishes there
x = 1 only, because only the positive square root is considered
x = −1 and x = 1, because f′(x) = 3x² − 3 equals zero at both
x = −3 and x = 3, because the coefficient of x is −3
11. Which statement best describes f′(a) when the derivative exists?
It is the instantaneous rate of change at x = a and the slope of the tangent line there.
It is the accumulated signed area between the graph and the x-axis from 0 to a.
It is the average value of the function over an interval containing a.
It is the x-coordinate at which the function reaches its greatest value.
12. Which derivative results from applying the chain rule to h(x) = e^(2x+1)?
e^(2x+1), with no factor from the exponent
e^(2x+1) + 2, with the inner derivative added
2e^(2x+1), with the inner derivative multiplied
(2x + 1)e^(2x), with the exponent treated as a product
13. A particle has position s(t) = t³ − 6t² + 9t. What is its instantaneous velocity at t = 2?
−4 units per time, from differentiating the quadratic term as −6t
−5 units per time, from using the position value s(2) as velocity
−3 units per time, from evaluating v(t) = 3t² − 12t + 9
3 units per time, from discarding the negative middle term
14. The curve y = x² − 4x has a tangent line at x = 3. What is the slope of that tangent line?
6, obtained from differentiating x² but not −4x
3, equal to the x-coordinate of the tangent point
5, equal to the function's value at x = 3
2, obtained by evaluating 2x − 4 at x = 3
15. Why does f(x) = |x| not have a derivative at x = 0?
The function is discontinuous at zero, so no tangent slope can be considered.
The function value is zero, which forces the derivative to be zero.
The right-hand slope is 1, so the derivative must equal 1.
The left-hand slope is −1 and the right-hand slope is 1, so the one-sided derivatives disagree.
16. For x > 0, which expression is the derivative of f(x) = ln(x)/x?
1/x², from differentiating the numerator and denominator separately
(1 − ln(x))/x², from applying the quotient rule
(x − ln(x))/x², from using 1 as the derivative of ln(x)
ln(x) − 1, from canceling the denominator after differentiation