Further Differentiation (A Level only) (OCR A Level Maths: Pure)

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.

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 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