Further Differentiation (DP IB Maths: AA HL)

Use the product rule to find the derivative of 

Use the quotient rule to find the derivative of 

Use the chain rule to find the derivative of 

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

Consider the function defined by 

By considering the derivative of the function, show that  is increasing everywhere on its domain.

Consider the function  defined by 

Show that the equation of the tangent to the graph of  at  may be written in the form 

Show that there is a point on the graph of at which the normal to the graph is vertical, and determine the coordinates of that point.

Consider the function defined by 

Find an expression for

Hence determine an equation for the tangent to the graph of  at  .

Let  where  and are functions such that  for all

Given that  and  find the equation of the tangent to the graph of  at 

Let be a function defined by 

Find an expression for

Determine the values of  for which the graph of  is

Your answers should be given as exact values.

Hence show that the graph of has two points of inflection, and determine the exact values of their coordinates.

Consider the function defined by  for 

Find the number of points at which the graph of has a horizontal tangent.

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

Show algebraically that the gradient of the tangent to the graph of at point A is 

Hence find the equation of the normal line to the graph of  at point A, and determine where that line intersects the -axis.

Show algebraically that the graph of  intersects the line in exactly three places, and determine the coordinates of the points of intersection.

Let  and  for 

Solve the equation 

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

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.