Differentiation (Edexcel A Level 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 .

A curve has equation

Find writing your answer in simplest form.

A curve has equation

Find writing your answer in simplest form.

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

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.

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 .

A curve has equation

where , and are constants.

• The gradient at the point is

• The gradient at the point is

Find , and .

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

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

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 .

A curve has equation

Show that

where and are constants to be found.

A different curve has equation

Show that

where and and are constants to be found.