# Differentiation

## Practise the technique of differentiating polynomials and other functions with this self marking exercise.

##### Menu  L 1L 2L 3L 4L 5L 6L 7L 8L 9Level 10L 11  Exam    Help

This is level 10: interpreting derivatives and second derivatives, maxima, minima and points of inflection. You can earn a trophy if you get at least 7 questions correct and you do this activity online.

You may use a scientific calculator but not a GDC for this exercise

 1. Find the second derivative of $$f$$ if $$f(x) = 8x^2 - 9x - 9$$. 2. If $$y =4x^4-bx^3+2x^2-3x$$find the value of $$b$$ if $$\quad \frac{d^2y}{dx^2} = 48x^2-12x+4$$. 3. If $$y =8x^4-5x^3+2x^2-7x$$find the value of the second derivative, $$\frac{d^2y}{dx^2}$$, when $$x=9$$ 4. A function $$f$$ has a derivative $$f'(x)=0$$ and a second derivative $$f''(x)=2.3$$ at the point where $$x= -1$$.Which of the following features does the function have at $$x= -1$$?a. A local minimumb. A local maximumc. A point of inflectionType in the letter that represents your answer 5. The graph of $$y=x^4-3x^3+5x$$ has a stationary point where $$x=1$$.By finding the first and second derivatives determine the nature of this stationary point.a. A local minimumb. A local maximumc. A point of inflectionType in the letter that represents your answer 6. Find the local minimum value of the function $$f$$ if$$\quad f(x) = \frac{x^3}{3} - \frac{x^2}{2} -30x+5$$ 7. If $$f(x) = e^x$$ what is the value of the second derivative of $$f(x)$$ when $$f(x)= 28.3$$? 8. If $$f(x) = \sin 2x$$ what is the value of the second derivative of $$f(x)$$ when $$f(x)= 0.68$$? 9. The function $$f$$ is defined by the formula $$f(x) = 4(6x-1)^2$$. By first finding the gradient function determine the value of x when the gradient is 7.Give your answer correct to three significant figures. 10. If $$y =3e^{3x}- \frac{2}{x^3}$$find the value of the second derivative, $$\frac{d^2y}{dx^2}$$, when $$x=3$$Give your answer correct to three significant figures.
Check

This is Differentiation level 10. You can also try:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8 Level 9 Level 11 Integration

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file.

## More Activities:

Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?

Comment recorded on the 24 May 'Starter of the Day' page by Ruth Seward, Hagley Park Sports College:

"Find the starters wonderful; students enjoy them and often want to use the idea generated by the starter in other parts of the lesson. Keep up the good work"

Comment recorded on the 19 October 'Starter of the Day' page by E Pollard, Huddersfield:

"I used this with my bottom set in year 9. To engage them I used their name and favorite football team (or pop group) instead of the school name. For homework, I asked each student to find a definition for the key words they had been given (once they had fun trying to guess the answer) and they presented their findings to the rest of the class the following day. They felt really special because the key words came from their own personal information."

Each month a newsletter is published containing details of the new additions to the Transum website and a new puzzle of the month.

The newsletter is then duplicated as a podcast which is available on the major delivery networks. You can listen to the podcast while you are commuting, exercising or relaxing.

Transum breaking news is available on Twitter @Transum and if that's not enough there is also a Transum Facebook page.

#### Where's Wallaby?

Find the hidden wallaby using the clues revealed at the chosen coordinates. Not only is this a fun way to practise using coordinates it is also a great introduction to Pythagoras' theorem and loci.

There are answers to this exercise but they are available in this space to teachers, tutors and parents who have logged in to their Transum subscription on this computer.

A Transum subscription unlocks the answers to the online exercises, quizzes and puzzles. It also provides the teacher with access to quality external links on each of the Transum Topic pages and the facility to add to the collection themselves.

Subscribers can manage class lists, lesson plans and assessment data in the Class Admin application and have access to reports of the Transum Trophies earned by class members.

Subscribe

## Go Maths

Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

## Maths Map

Are you looking for something specific? An exercise to supplement the topic you are studying at school at the moment perhaps. Navigate using our Maths Map to find exercises, puzzles and Maths lesson starters grouped by topic.

## Teachers

If you found this activity useful don't forget to record it in your scheme of work or learning management system. The short URL, ready to be copied and pasted, is as follows:

Alternatively, if you use Google Classroom, all you have to do is click on the green icon below in order to add this activity to one of your classes.

It may be worth remembering that if Transum.org should go offline for whatever reason, there is a mirror site at Transum.info that contains most of the resources that are available here on Transum.org.

When planning to use technology in your lesson always have a plan B!

Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.

For Students:

For All:

© Transum Mathematics :: This activity can be found online at:
www.Transum.org/go/?Num=55

## Description of Levels

Close

Before beginning these exercises make sure you understand Indices really well.

Level 1 - Differentiate basic polynomials

Level 2 - Differentiate polynomials including negative and fractional indices

Level 3 - Calculations involving the gradient at the given point

Level 4 - Finding tangents and normals

Level 5 - Differentiate trigonometric functions

Level 6 - Differentiate exponential and natural logarithm functions

Level 7 - Differentiate using the chain rule

Level 8 - Differentiate using the product rule

Level 9 - Differentiate using the quotient rule

Level 10 - Interpreting derivatives and second derivatives, maxima, minima and points of inflection.

Level 11 - Differentiate simple functions parametrically

Exam Style questions are in the style of IB or A-level exam paper questions and worked solutions are available for Transum subscribers.

Integration - Exercises on indefinite and definite integration of basic algebraic and trigonometric functions.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

## Terminology and symbols

Please note that if $$y = f(x) = x^2$$ then the first differential can be shown in any of the following ways:

$$\frac{dy}{dx} = 2x$$ $$y' = 2x$$ $$f'(x) = 2x$$

## Differentiating Trigonometric Functions

$$\frac{d}{dx} (\sin x) = \cos x$$ $$\frac{d}{dx} (\cos x) = -\sin x$$ $$\frac{d}{dx} (\tan x) = \frac{1}{\cos^2 x}$$

## Differentiating Other Functions

$$\frac{d}{dx} (e^x) = e^x$$ $$\frac{d}{dx} ( \ln x) = \frac{1}{x}$$

In the following rules, $$u$$ and $$v$$ are functions of $$x$$.

## The Product Rule

$$\text{If} \quad y = uv \quad \text{then}$$ $$\frac{dy}{dx} = v\frac{du}{dx} + u\frac{dv}{dx}$$

## The Quotient Rule

$$\text{If} \quad y = \frac{u}{v} \quad \text{then}$$ $$\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$$

## The Chain Rule

$$\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$$

## Parametric Equations

if $$x$$ and $$y$$ are given in terms of a third variable, the parameter, which could be $$t$$, then:

$$\frac{dy}{dx} = \frac{dy}{dt} \div \frac{dx}{dt}$$