Chapter Review Derivative Rules
Chapter
Chapter 2
Section
Chapter Review Derivative Rules
Solutions 69 Videos

Use the definition of the derivative to find f'(x) for each of the following functions.

y=2x^2-5x

1.38mins
Q2a

Find f'(x) using the definition of derivative.

 \displaystyle y = \sqrt{x-6} 

Q2b

Differentiate each of the following functions:

y=x^2-5x+4

0.14mins
Q3a

Differentiate each of the following functions:

\displaystyle{f(x)=x^{\displaystyle{\frac{3}{4}}}}

0.27mins
Q3b

Differentiate each of the following functions:

\displaystyle{y=\frac{1}{x^2+5}}

0.23mins
Q3d

Differentiate each of the following functions:

\displaystyle{y=\frac{3}{(3-x^2)^2}}

0.51mins
Q3e

Differentiate each of the following functions:

\sqrt{7x^2+4x+1}

0.47mins
Q3f

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

f(x)=\displaystyle{\frac{2x^3-1}{x^2}}

0.30mins
Q4a

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

g(x)=\sqrt{x}(x^3-x)

0.18mins
Q4b

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

\displaystyle{y=\frac{x}{3x-5}}

0.25mins
Q4c

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

y=\sqrt{x-1}(x+1)

0.41mins
Q4d

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

f(x)=(\sqrt{x}+2)^{\displaystyle{\frac{-2}{3}}}

0.39mins
Q4e

Determine the derivative of the given function. In some cases, it will save time if you rearrange the function before differentiating.

\displaystyle{y=\frac{x^2+5x+4}{x+4}}

0.24mins
Q4f

y=x^4(2x-5)^6

1.18mins
Q5a

y=x\sqrt{x^2+1}

0.52mins
Q5b

y=\displaystyle{\frac{(2x-5)^4}{(x+1)^3}}

1.42mins
Q5c

y=\displaystyle{\left(\frac{10x-1}{3x+5} \right)^6}

1.21mins
Q5d

y=(x-2)^3(x^2+9)^4

1.21mins
Q5e

y=(1-x^2)^3(6+2x)^{-3}

2.06mins
Q5f

If f is a differentiable function, find an expression for the derivative of each of the following functions:

g(x)=f(x^2)

0.10mins
Q6a

If f is a differentiable function, find an expression for the derivative of each of the following functions:

h(x)=2xf(x)

0.27mins
Q6b

If y=5u^2+3u-1 and u=\displaystyle{\frac{18}{x^2+5}}, find \displaystyle{\frac{dy}{dx}} when x=2.

1.32mins
Q7a

If y=\displaystyle{\frac{u+4}{u-4}} and u=\displaystyle{\frac{\sqrt{x}+x}{10}}, find \displaystyle{\frac{dy}{dx}} when x=4.

1.57mins
Q7b

If y=f(\sqrt{x^2+9}) and f'(5)=-2, find \displaystyle{\frac{dy}{dx}} when x=4.

1.12mins
Q7c

Determine the slope of the tangent at point (1,4) on the graph of f(x)=(9-x^2)^{\displaystyle{\frac{2}{3}}}.

0.42mins
Q8

a. For what values of x does the curve y=-x^3+6x^2 have a slope of -12?

b. For what values of x does the curve y=-x^3+6x^2 have a slope of -15?

1.32mins
Q9

Determine the values of x where the graph of each function has a horizontal tangent.

i. y=(x^2-4)^5

ii.  \displaystyle y = (x^3 - x)^2 

1.18mins
Q10a

Determine the equation of the tangent to each function at the given point.

y=(x^2+5x+2)^4, (0,16)

0.43mins
Q11a

Determine the equation of the tangent to each function at the given point.

y=(3x^{-2}-2x^3)^5, (1,1)

0.46mins
Q11b

A tangent to the parabola y=3x^2-7x+5 is perpendicular to x+5y-10=0. Determine the equation of the tangent.

1.17mins
Q12

The line y=8x+b is tangent to the curve y=2x^2. Determine the point of tangency and the value of b.

0.45mins
Q13

Consider the function f(x)=2x^{\displaystyle{\frac{5}{3}}}-5x^{\displaystyle{\frac{2}{3}}}

Determine the slope of the tangent at the point where the graph crosses the x-axis.

1.44mins
Q15a

Consider the function f(x)=2x^{\displaystyle{\frac{5}{3}}}-5x^{\displaystyle{\frac{2}{3}}}

• Determine the value of a shown in the graph of f(x) given below.
0.21mins
Q15b

A rested student is able to memorize M words after t minutes, where M=0.1t^2-0.001t^3, 0\leq t\leq \displaystyle{\frac{200}{3}}.

i. How many words are memorized in the first 10 min?

ii. How many words are memorized in the first 15 min?

0.53mins
Q16a

A rested student is able to memorise M words after t minutes, where M=0.1t^2-0.001t^3, 0\leq t\leq \displaystyle{\frac{200}{3}}.

i. What is the memory rate at t=10?

ii. What is the memory rate at t=15?

1.08mins
Q16b

A grocery store determines that, after t hours on the job, a new cashier can scan N(t)=20-\displaystyle{\frac{30}{\sqrt{9+t^2}}} items per minute.

a) Find N'(t), the rate at which the cashier's productivity is changing.

b) According to this model, does the cashier ever stop improving? Why?

1.48mins
Q17

An athletic-equipment supplier experiences weekly costs of C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700in producing x baseball gloves per week.

Find the marginal cost, C'(x)

0.19mins
Q18

An athletic-equipment supplier experiences weekly costs of C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700in producing x baseball gloves per week.

• Find the production level x at which the marginal cost is \$76 per glove.
0.24mins
Q18b

A manufacturer of kitchen appliances experiences revenue of R(x)750x-\displaystyle{\frac{x^2}{6}}-\displaystyle{\frac{2}{3}}x^3 dollars from the sale of x refrigerators per month.

• Find the marginal revenue, R'(x).
0.25mins
Q19a

A manufacturer of kitchen appliances experiences revenue of R(x)750x-\displaystyle{\frac{x^2}{6}}-\displaystyle{\frac{2}{3}}x^3 dollars from the sale of x refrigerators per month.

• Find the marginal revenue when 10 refrigerators per month are sold.
0.21mins
Q19b

An economist has found that the demand function for a particular new product is given by D(p)=\displaystyle{\frac{20}{\sqrt{p-1}}}, p>1. Find the slope of the demand curve at the point (5,10).

1.14mins
Q20

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

• Find B(0) and B(30).
0.27mins
Q21a

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

• Find B'(0) and B'(30)
0.28mins
Q21b

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

• Find B'(0) and B'(30) and interpret your results.
0.21mins
Q21c

Kathy has diabetes. Her blood sugar level, B, one hour after an insulin injection, depends on the amount of insulin, x in milligrams injected. B(x)=-0.2x^2+500, 0\leq x\leq 40.

• Find the values of B'(50) and B(50).
1.01mins
Q21d

Determine if the function is differentiable at x=1.

m(x)=|3x-3|-1

1.19mins
Q22d

At what x-values is each function not differentiable? Explain.

f(x)=\displaystyle{\frac{3}{4x^2-x}}

0.17mins
Q23a

At what x-values is each function not differentiable? Explain.

f(x)=\displaystyle{\frac{x^2-x-6}{x^2-9}}

0.31mins
Q23b

At what x-values is each function not differentiable? Explain.

f(x)=\sqrt{x^2-7x+6}

0.50mins
Q23c

At a manufacturing plant, productivity is measured by the number of items, p, produced per employee per day over the previous 10 years. Productivity is modelled by p(t)=\displaystyle{\frac{25t}{t+1}}, where t is the number of years measured from 10 years ago. Determine the rate of change of p with respect to t.

0.32mins
Q24

Given f(x)=\frac{(2x-3)^2+5}{2x-3},

Express f as the composition of two simpler functions.

0.32mins
Q26a

Given f(x)=\frac{(2x-3)^2+5}{2x-3} and y = u+5u^{-1}, u = 2x -3

Use this composition to determine f'(x)

0.50mins
Q26b

Given g(x)=\sqrt{2x-3}+5(2x-3),

• Express g as the composition of two simpler functions.
0.39mins
Q27a

Given g(x)=\sqrt{2x-3}+5(2x-3),

Use this composition to determine g'(x).

0.28mins
Q27b

Determine the derivative of the function.

f(x)=(2x-5)^3(3x^2+4)^5

1.46mins
Q28a

Determine the derivative of the function.

g(x)=(8x^3)(4x^2+2x-3)^5

1.52mins
Q28b

Determine the derivative of the function.

y=(5+x)(4-7x^3)^6

2.25mins
Q28c

Determine the derivative of the function.

h(x) = \frac{6x - 1}{(3x + 5)^4}

1.27mins
Q28d

Determine the derivative of the function.

y=\displaystyle{\frac{(2x^2-5)}{(x+8)^2}}

1.29mins
Q28e

Determine the derivative of the function.

f(x)=\displaystyle{\frac{-3x^4}{\sqrt{4x-8}}}

2.09mins
Q28f

Determine the derivative of the function.

g(x)=\displaystyle{\left( \frac{2x+5}{6-x^2} \right)^4}

1.51mins
Q28g

Determine the derivative of the function.

y=\displaystyle{\left[ \frac{1}{(4x+x^2)^3}\right]^3}

0.45mins
Q28h

Find numbers a, b, and c so that the graph f(x)=ax^2+bx+c has x-intercepts at (0,0) and (8,0), and a tangent with slope 16 where x=2.

1.24mins
Q29

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

Find A'(t).

0.07mins
Q30a

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

Find the rate of change of the number of living ants in the colony at 5 h.

0.17mins
Q30b

An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.

• How many ants were in the colony before it was treated with the insecticide?
An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t)=-t^3+5t+750.