Further Differentiation (Edexcel A Level Maths: Pure): Exam Questions

Given that

show that

where is a rational constant to be found.

Figure 1 shows a sketch of the curve with equation

Show that

The point , shown in Figure 1, is the minimum turning point on .

Show that the coordinate of is a solution of

A car stops at two sets of traffic lights.

Figure 2 shows a graph of the speed of the car, , as it travels between the two sets of traffic lights.

The car takes seconds to travel between the two sets of traffic lights.

The speed of the car is modelled by the equation

where seconds is the time after the car leaves the first set of traffic lights.

According to the model, find the value of

Show that the maximum speed of the car occurs when

Given that is measured in radians, prove, from first principles, that

You may assume the formula for and that as , and

The function is defined by

where is a positive constant.

Show that

where is a function to be found.

Given that the curve with equation has at least one stationary point, find the range of possible values of .

where is measured in radians.

Use differentiation from first principles to show that

You may

• use without proof the formula for

• assume that as , and

The function is defined by

where is a constant.

Deduce the value of .

Prove that

for all values of in the domain of .

Given that

Show that

Hence prove that .

A curve has the equation 

Find the gradient of the normal to the curve at the point , giving your answer correct to 3 decimal places.

Find    for each of the following:

Find   for each of the following:

Find the equation of the tangent to the curve    at the point , giving your answer in the form, where a, b and c are integers.

Differentiate with respect to x, simplifying your answers as far as possible:

Differentiate  with respect to x.

Show that if  , then 

Hence find the gradient of the tangent to the curve  at the point with coordinates 

The diagram below shows part of the graph of  , whereis the function defined by

Points A, B and C are the three places where the graph intercepts the x-axis.

Find .

Show that the coordinates of point A are (-2, 0).

Find the equation of the tangent to the curve at point A.

Figure 2 shows a sketch of the curve with equation where

Show that .

Hence find, in simplest form, the exact coordinates of the stationary points of .

The function and the function are defined by

Find

(i) the range of

(ii) the range of .

Find the values of the constants , and .

Prove that is a decreasing function.

Given that

show that

where is a constant to be found.

A curve has equation , where

Show that

where and are constants to be found.

Hence show that the coordinates of the turning points of the curve are solutions of the equation

Show from first principles that the derivative of  is .

A curve has the equation  

Show that the equation of the tangent to the curve at the point with x-coordinate 1 is

For  , where is a real number and is an integer, show that

Find the gradient of the normal to the curve at the point with x-coordinate 0.  Give your answer correct to 3 decimal places. 

Differentiate with respect to x, simplifying your answers as far as possible:

By writing    as  and then using the product and chain rules, show that

Given that ,

Find    in terms of y

Hence find in terms of x.

The diagram below shows part of the graph of , where is the function defined by

Point A is a maximum point on the graph.

Show that the x-coordinate of A is a solution to the equation

Show that the -coordinates of the turning points of the curve with equation satisfy the equation

Figure 3 shows a sketch of part of the curve with equation .

Sketch the graph of against where

showing the long-term behaviour of this curve.

The function is used to model the height, in metres, of a ball above the ground seconds after it has been kicked.

Using this model, find the maximum height of the ball above the ground between the first and second bounce.

The curve , in the standard Cartesian plane, is defined by the equation

The curve passes through the origin

Find the value of at the origin.

(i) Use the small angle approximation for to find an equation linking and for points close to the origin.

(ii) Explain the relationship between the answers to (a) and (b)(i).

Show that, for all points lying on ,

where and are constants to be found.

A scientist is studying a population of mice on an island.

The number of mice, , in the population, months after the start of the study, is modelled by the equation

Find the number of mice in the population at the start of the study.

Show that the rate of growth is given by

The rate of growth is a maximum after months.

Find, according to the model, the value of .

According to the model, the maximum number of mice on the island is .

State the value of .

Show from first principles that the derivative of is .

A curve has the equation .

Show that the gradient of the normal to the curve at the point  is

Find the derivative of the function 

Show that the derivative    is

Hence find the equation of the tangent to the curve at the point , giving your answer in the form , where a and b are to be given as exact values.

Differentiate with respect to x, simplifying your answers where possible:

The diagram below shows the graph of , where is the function defined by

The points A and B are maximum and minimum points, respectively.

Find the range of , giving your answer correct to 3 decimal places.

A is the point on the graph of  such that the tangent to the graph at passes through the point . 
Show that the x-coordinate of A satisfies the equation

A sequence of functions is defined by the recurrence relation

Based on that sequence, the functionis defined by

Calculate the value of