Differentiation (Edexcel AS Maths: Pure): Exam Questions

Given that

use differentiation from first principles to show that

A curve has equation

Find writing your answer in simplest form.

The function is given by

Show that can be written in the form

where , , and are constants to be found.

Hence find an expression for .

Find the coordinates of any points on the curve

at which the gradient is zero.

Find the -coordinate of the point on the curve with equation

at which the gradient is 4.

A student uses differentiation from first principles to show that the derivative of is .

• Their working is shown below.

• There is an error in their argument in step 6.

Write out the correct argument for step 6.

STEP 1

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

When  , 

Given that

Use differentiation from first principles to show that

Given that

Use differentiation from first principles to show that

Given that

find .

Given that

find .

A curve has equation

Find writing your answer in simplest form.

A curve has equation

Find writing your answer in simplest form.

The function is defined by

Find writing your answer in simplest form.

Hence solve the equation

A curve has the equation

Find .

Find the coordinates of the point on the curve where the gradient is 2.

The function is defined by

where is a constant.

Find an expression for in terms of and .

Given that the equation has exactly one real solution, find the value of .

A curve has equation

Write the equation of the curve in the form

where , , and are constants to be found.

Hence find writing your answer in simplest form.

Find the coordinates of the point(s) on the curve where the gradient is .

A curve has the equation

where , and are non-zero constants.

• The gradient at the point is

• The gradient at the point is

Show that

and

Hence find and .

By considering the point , find .

A curve has equation

Find writing your answer in simplest form.

A curve has equation

Find .

Given that

Use differentiation from first principles to show that

where is a constant.

Given that

Use differentiation from first principles to show that

A curve has equation

Show that

where , , and are constants to be found.

A curve has equation

Find writing your answer in simplest form.

A curve has equation

Find .

The function is defined by

Show that there are no solutions to the equation

A curve has the equation

where is a constant.

There is only one point on the curve where the gradient of the tangent at that point is zero.

Find the possible values of .

The function is defined by

where is a constant.

Find all the possible values of for which the equation

has at least one real solution.

A curve has the equation

Show that 

where and are rational numbers to be found.

Hence find the coordinates of the point on the curve where the gradient is 0.

Given that

show that

where is a constant to be found.

A curve has equation

Find , writing your answer in the form

where

• , and are constants to be found

• , and are terms of a descending arithmetic sequence to be found

A curve has equation 

Find , writing the terms in your answer in ascending powers of .

A curve has the equation

where , and are non-zero constants.

• The curve passes through the point

• The gradient at the point is

Find , and .