5.2 Further Differentiation

Find  for each of the following:

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

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

Let  where  and 

Find the equation of the tangent of  at 

A curve has the equation 

Find expressions for  and .

Determine the coordinates of the local minimum of the curve.

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

Points  and are the three places where the graph intercepts the  -axis.

Find 

Show that the coordinates of point are

Find the equation of the tangent to the curve at point 

Let 

Find 

Find 

Find the exact  of the points of inflection for the graph of .

Find 

Let  where 

Find the number of points containing a horizontal tangent.

Show algebraically that the gradient of the tangent at  is 

State the gradient of the tangent at 

It can be found that as the function,  undergoes a transformation  the number of stationary points found between   increases.

Find the number of stationary points on  after a transformation of and hence, state the general rule representing the number of stationary points in terms of  where

Let  and  for 

Solve 

Find an expression for the derivative of each of the following functions:

Consider the function  defined by   

Show that  is decreasing everywhere on its domain.

Consider the function  defined by 

Point A is the point on the graph of  for which the -coordinate is 

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

Point B is the point on the graph of  at which the normal to the graph is vertical.

Show that the coordinates of the point of intersection between the tangent to the graph of  at point A and the tangent to the graph of  at point B are

Consider the function  defined by 

Show that the normal line to the graph of  at intercepts the -axis at the point

Let  , where and are real-valued functions such that

for all 

Given that  and , where , find the distance between the -intercept of the tangent to the graph of at  and the -intercept of the normal to the graph of at . Give your answer in terms of and/or as appropriate.

Consider the function  defined by  where  is a positive integer.

For the case where , find the number of points in the interval   at which the graph of  has a horizontal tangent.

Show algebraically that in general the -coordinates of the points at which the graph of  has horizontal tangents will be the solutions to the equation

Hence, for the case where , find the -coordinates of the points identified in part (a).

In terms of , state in general how many (i) turning points and (ii) points of inflection the graph of  will have in the interval  Give a reason for your answers.

Let  where  and are well-defined functions with  anywhere on their common domain.

By first writing  , use the product and chain rules to show that

Consider the function  defined by  where is a positive integer.

Show that the graph of  will have no points of inflection in the case where .

Show that, for  the second derivative of  is given by

Explain why, for 

Hence show that the graph of  will only have points of inflection in the case where is an odd integer greater than or equal to 3. In that case, give the exact coordinates of the points of inflection, giving your answer in terms of  where appropriate.