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Solutions
69 Videos

Use the definition of the derivative to find `f'(x)`

for each of the following functions.

`y=2x^2-5x`

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1.38mins

Q2a

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

```
\displaystyle
y = \sqrt{x-6}
```

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Q2b

Differentiate each of the following functions:

`y=x^2-5x+4`

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0.14mins

Q3a

Differentiate each of the following functions:

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

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0.27mins

Q3b

Differentiate each of the following functions:

`\displaystyle{y=\frac{7}{3x^4}}`

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0.28mins

Q3c

Differentiate each of the following functions:

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

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0.23mins

Q3d

Differentiate each of the following functions:

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

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0.51mins

Q3e

Differentiate each of the following functions:

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

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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}}`

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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)`

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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}}`

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0.25mins

Q4c

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

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0.41mins

Q4d

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

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0.39mins

Q4e

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

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0.24mins

Q4f

Determine the derivative, and give your answer in a simplified form.

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

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1.18mins

Q5a

Determine the derivative, and give your answer in a simplified form.

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

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0.52mins

Q5b

Determine the derivative, and give your answer in a simplified form.

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

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1.42mins

Q5c

Determine the derivative, and give your answer in a simplified form.

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

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1.21mins

Q5d

Determine the derivative, and give your answer in a simplified form.

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

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1.21mins

Q5e

Determine the derivative, and give your answer in a simplified form.

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

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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)`

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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)`

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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`

.

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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`

.

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1.57mins

Q7b

If `y=f(\sqrt{x^2+9})`

and `f'(5)=-2`

, find `\displaystyle{\frac{dy}{dx}}`

when `x=4`

.

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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}}}`

.

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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`

?

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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
```

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1.18mins

Q10a

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

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

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0.43mins

Q11a

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

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

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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.

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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`

.

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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.

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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.

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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?

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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`

?

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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?

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1.48mins

Q17

An athletic-equipment supplier experiences weekly costs of `C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700`

in producing `x`

baseball gloves per week.

Find the marginal cost, `C'(x)`

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0.19mins

Q18

An athletic-equipment supplier experiences weekly costs of `C(x)=\displaystyle{\frac{1}{3}}x^3+40x+700`

in producing `x`

baseball gloves per week.

- Find the production level
`x`

at which the marginal cost is $76 per glove.

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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)`

.

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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.

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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)`

.

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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)`

.

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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)`

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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.

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0.21mins

Q21c

`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)`

.

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1.01mins

Q21d

Determine if the function is differentiable at `x=1`

.

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

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1.19mins

Q22d

At what `x`

-values is each function not differentiable? Explain.

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

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0.17mins

Q23a

At what `x`

-values is each function not differentiable? Explain.

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

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0.31mins

Q23b

At what `x`

-values is each function not differentiable? Explain.

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

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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`

.

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0.32mins

Q24

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

,

Express `f`

as the composition of two simpler functions.

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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)`

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0.50mins

Q26b

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

,

- Express
`g`

as the composition of two simpler functions.

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0.39mins

Q27a

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

,

Use this composition to determine `g'(x)`

.

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0.28mins

Q27b

Determine the derivative of the function.

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

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1.46mins

Q28a

Determine the derivative of the function.

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

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1.52mins

Q28b

Determine the derivative of the function.

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

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2.25mins

Q28c

Determine the derivative of the function.

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

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1.27mins

Q28d

Determine the derivative of the function.

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

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1.29mins

Q28e

Determine the derivative of the function.

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

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2.09mins

Q28f

Determine the derivative of the function.

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

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1.51mins

Q28g

Determine the derivative of the function.

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

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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`

.

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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)`

.

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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.

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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?

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0.10mins

Q30c

`A`

, in hundreds at `t`

hours is `A(t)=-t^3+5t+750`

.

How many hours after the insecticide was applied were no ants remaining in the colony?

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0.31mins

Q30d