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Core 2: Differentiation
This section covers the key skills
required to be able to answer the questions on ASlevel differentiation. If you
are looking for notes on chainrule/productrule/quotient rule then look under
Core 3 differentiation.
View our playlist for differentiation

View our Video on differentiating more complex powers

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 straightline 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 straightline we can use the general equation: . The
only value we still don’t know is the ycoordinate 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 Rearrange 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 ycoordinates. The
general rule is to differentiate again. Into
this new function put the xvalues 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. 