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Core 2: Differentiation

This section covers the key skills required to be able to answer the questions on AS-level differentiation. If you are looking for notes on chain-rule/product-rule/quotient rule then look under Core 3 differentiation.

General form for differentiation

 

If then

 

Differentiating finds the gradient functions. In other words it tells you the gradient for a given value of x.

The general rule can be expressed in words as:

“Multiply by the power and decrease the power by one”

Question

 

 

Answer

 

 

 

 

This process works for any power. I suggest you revise and learn all the following more awkward powers

 

 

Question and hint

 

Answer

 

This is the same as,

 

 

 

This can be written as;

 

 

 

This is the same as,

 

 

This can be written as,

 

 

 

This is the same as,

 

 

 

This can be written as,

 

 

You must also be able to cope with expressions which need the powers to be initially simplified. You must draw on all the indices skills met at GCSE and during core 1.

 

 

Questions and hint

 

Answers

This is the same as:

 

 

 

 

 

Remember that when you multiply, you add the powers.

 

Now differentiate,

 

 

Expand:

 

 

 

 

 

Now differentiate:

 

 

 

You must be able to apply these skills in a range of contexts. The main two are:

1)           Finding the equations of tangents and normal lines

2)          Finding and classifying turning points/stationary points

 

Tangents and normal lines

 

A curve is given by:

 

 

Find the equation of the tangent at the point x=2

 

 

At x=2:

 

So m=20

 

 

 

 

 

 

 

 

 

 

 

So, the equation is:

 

 

Find the equation of the normal when x=2

 

The normal is perpendicular to the tangent, so the gradient is given by:

 

So,

 

 

 

 

 

Explanation

 

 

 

 

 

The tangent is a straight-line which shares the same gradient as the curve at the point where it touches. So we need to differentiate to find the gradient.

 

 

 

 

 

Because the tangent is a straight-line we can use the general equation:

.

 

The only value we still don’t know is the y-coordinate when x=2. We can find this using:

 

 

 

 

 

 

 

 

 

 

To find a perpendicular gradient, reciprocate the known gradient and change the sign.

 

 

 

 

 

Multiply by 20

 

Expand the bracket

 

Re-arrange

 

Final answer

Turning points, also known as stationary points.

 

 

 

 

Find the turning points of:

 

 

Well,

Need,

 

 

 

So

 

Classify the turning points

 

Second differential

 

So this is a minimum point

 

So this is a maximum point

 

These are defined by the points which satisfy the property:

 

 

 

 

 

 

 

 

 

Divide by 3

 

Factorise

 

 

 

Find the corresponding y-coordinates.

 

The general rule is to differentiate again.

Into this new function put the x-values of the turning points.

Then:

1)           If answer is positive you have a minimum point

2)          If answer is negative you have a maximum point

3)          If answer is zero it is probably a point of inflection, though it could be one of the above. In this case you need to analyse the sign of the gradient either side of turning point.