Further Differentiation (CIE A Level Maths: Pure 3)

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 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.

Use the chain rule to differentiate   ,  where  is a real constant.

What does your answer to part (a) tell you about the number of stationary points on the curve 

Use an appropriate method to differentiate each of the following.

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

Use the chain rule to show that the derivative of , where  is a real constant, is a

Hence find the coordinates of any stationary point(s) on the curve with equation

giving your answers as exact values.

Use an appropriate method to differentiate each of the following.

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 sequence of functions is defined by the recurrence relation

Based on that sequence, the functionis defined by

Calculate the value of 

Use calculus to find the coordinates of the stationary points of the curve

and determine whether each one is a maximum or a minimum.  The coordinates should be given as exact values.