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Differentiation

Practise the technique of differentiating polynomials and other functions with this self marking exercise.
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This is level 4: finding tangents and normals. You can earn a trophy if you get at least 9 questions correct and you do this activity online. Write your answer in the form \(y=mx+c\) for the first seven questions.

Find the equation of the tangent to the curve
\(y=x^2-3\)
at the point (2, 1).

Correct Wrong

Find the equation of the tangent to the curve \(y=5x^2+7\)
at the point (1,12).

Correct Wrong

Find the equation of the tangent to the curve \(y=x^3-15x\)
at the point (3,-18).

Correct Wrong

Find the equation of the tangent to the curve \(y=x^2+2\)
where \(x = 1\).

Correct Wrong

Find the equation of the tangent to the curve \(y=4x^2+8\)
where \(x = -1\).

Correct Wrong

Find the equation of the tangent to the curve \(y=1-x^2\)
where \(x = -3\).

Correct Wrong

Find the equation of the normal to the curve
\(y=x^2-7\)
at the point (2,-3).
Write your answer in the form \(x+by+c=0\) where \(b\) and \(c\) are integers.

Correct Wrong

Find the equation of the normal to the curve \(y=10-x^2\)
where x = 3.
Write your answer in the form \(by=x+c\) where \(b\) and \(c\) are integers.

Correct Wrong

Find the equation of the normal to the curve \(y=x^3-7x+2\)
where x = -1.
Write your answer in the form \(by=x+c\) where \(b\) and \(c\) are integers.

Correct Wrong

Find the equation of the normal to the curve \(y=3x^2-5x-1\)
where \(x=1\).

Correct Wrong

Find the equation of the normal to the curve \(y=4-x^2\) where the x-coordinate is 0.

Correct Wrong

Find the equation of the tangent to the curve \(y=x^2\) which is parallel to the x-axis.

Correct Wrong

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This is Differentiation level 4. You can also try:
Level 1 Level 2 Level 3 Level 5 Level 6 Level 7 Level 8 Level 9 Level 10 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.
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Did you enjoy doing this 'Differentiation' activity? Are you curious about who originally came up with this idea in Maths? Discover more about one of the mathematicians who is associated with this concept.
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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:

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

Richard, Pennsylvania

Sunday, October 9, 2022

"Level 7:
y= sqrt(5x^2 -8x)
The answer is (5x-4)/sqrt(5x^2-8x) but for some reason it is marked as wrong. I want to confirm that this isn't an error or anything. Thanks!

[Transum: Thanks for your feedback Richard. You are right. The software stores a number of different versions of the correct answer and the format you chose was missing. I have now corrected this error. Sorry for the inconvenience and thanks again for taking the time to let me know.]"

Mandy Fox,

Saturday, July 1, 2023

"Hi John, Please could we have a button to get from the Differentiation page to the Integration page. Many thanks! Mandy.

[Transum: Hi Mandy, thanks for the suggestion. there is now a button linking directly to the Integration exercise - there is also a link in the Menu tab.]"

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.

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Description of Levels

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

You should also have been shown how to differentiate from first principles.

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.

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Example

Mathematical Notation

Use the ^ key to type in a power or index then the right arrow or tab key to end the power.

For example: Type 3x^2 to get 3x2.

Use the forward slash / to type a fraction then the right arrow or tab key to end the fraction.

For example: Type 1/2 to get ½.

Fractions should be given in their lowest terms.

A square root sign (if required) should be typed in as \sqrt space then press the right arrow key after typing in the last term in the square root.

Typing Mathematical Notation

These exercises use MathQuill, a web formula editor designed to make typing Maths easy and beautiful. Watch the animation below to see how common mathematical notation can be created using your keyboard.

MathQuill Animation

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

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

Answer format

There are many ways you could correctly type in the answers that have a number of terms. The software in this page should recognise most of the commonly-used formats but if you are convinced you have the correct answer but it is being shown as incorrect try typing the answer in a different format. As always, check with your teacher if you are unsure.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

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.

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