01
True or false
1 point
True or false: At x = a, the derivative f'(a) represents both the instantaneous rate of change of f and the slope of the tangent line to its graph at x = a.
Free Practice Quiz Question List
05 Differentiation Techniques and Related Rates Online Quiz Questions
Use this free practice quiz with 20 questions to review 05 Differentiation Techniques and Related Rates, test your knowledge, and prepare for your next test or exam.
20 questions
02
Choose one
1 point
Find dy/dx for y = (x^2 + 1)/(x - 3).
- A
(x^2-6x+1)/(x-3)^2
- B
(x^2-6x-1)/(x-3)^2
- C
(2x-1)/(x-3)
- D
(x^2+1)/(x-3)^2
03
Fill in the blank
1 point
Using the chain rule, find dy/dx for y = (3x^2 - 5)^4. Enter your result:
04
Written response
1 point
Suppose f(2) = 5 and f'(2) = 3. What is the exact value of (f^(-1))'(5)?
05
Choose all
1 point
Select all statements that correctly describe how to apply the chain rule to a composite function.
- A
For f(g(x)), the derivative is f'(g(x))g'(x).
- B
For a composition, differentiate only the innermost function.
- C
In a multistep composition, work through the layers from the outside inward.
- D
The inner function remains unchanged while the outer function is differentiated at that step.
- E
The chain rule says the derivative of a composition is always the product of the two original functions.
06
True or false
1 point
True or false: When differentiating an implicitly defined equation with respect to x, d/dx(y^2) must be written as 2y(dy/dx).
- A
True
- B
False
07
Choose one
1 point
The circle x^2 + y^2 = 25 contains the point (3, 4). What is the slope dy/dx at that point?
- A
3/4
- B
-4/3
- C
-3/4
- D
4/3
08
Fill in the blank
1 point
A spherical balloon has dV/dt = 2 cm^3/s. How fast is its radius increasing when r = 3 cm? Enter the exact value in cm/s:
09
Choose all
1 point
Let s(t) be a position function. Select all correct statements about its higher-order derivatives.
- A
s'(t) is velocity.
- B
s''(t) is acceleration.
- C
s''(t) is always the position.
- D
s'''(t) describes the rate of change of acceleration.
- E
Higher-order derivatives are obtained by differentiating the preceding derivative.
10
Choose one
1 point
For -1 < x < 1, which expression equals d/dx(arcsin x)?
- A
-1/sqrt(1-x^2)
- B
1/sqrt(1-x^2)
- C
sqrt(1-x^2)
- D
1/(1-x^2)
11
Open ended
1 point
Use implicit differentiation to find dy/dx for x^3 sin y + y = 4x + 3. Show how the product rule and chain rule are used, and simplify your final expression.
12
Choose one
1 point
If f(x) = x^5 - 3x^3 + 2x, what is f^(4)(x)?
- A
120x^2
- B
120x
- C
24
- D
0
13
True or false
1 point
True or false: The derivative of every constant function is zero.
- A
True
- B
False
14
Choose one
1 point
Which expression is the derivative of y = (x² + 1) sin x?
- A
2x cos x + (x² + 1) sin x
- B
2x sin x + (x² + 1) cos x
- C
2x sin x + cos x
- D
(2x + 1) cos x
15
Choose one
1 point
For x ≠ 3, which expression is the derivative of y = (x² + 1)/(x − 3)?
- A
(x² − 6x + 1)/(x − 3)²
- B
(x² − 6x + 1)/(x − 3)
- C
(x² − 6x − 1)/(x − 3)²
- D
(2x − 1)/(x − 3)²
16
Written response
1 point
Suppose f(2) = 5 and f'(2) = 3. What is the exact value of (f⁻¹)'(5)?
17
Choose one
1 point
Which expression is the derivative of y = (3x² − 5)⁴?
- A
24x(3x² − 5)³
- B
4(3x² − 5)³
- C
24(3x² − 5)³
- D
12x(3x² − 5)⁴
18
Choose one
1 point
The circle x² + y² = 25 contains the point (3, 4). What is the slope dy/dx at that point?
- A
3/4
- B
−4/3
- C
4/3
- D
−3/4
19
Written response
1 point
A 10-ft ladder leans against a wall. The bottom is 6 ft from the wall and is sliding away from the wall at 2 ft/s. At that instant, how fast is the top moving vertically? Enter the numerical value of dy/dt in ft/s as a simplified fraction.
20
Written response
1 point
A circular oil slick has area increasing at 6π m²/s. How fast is its radius increasing when the radius is 2 m? Enter the numerical value of dr/dt in m/s as a simplified fraction.