Further Differentiation (DP IB Maths: AI 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 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  .

By considering  show that there is a point on the graph of  at which the normal to the graph is vertical, and determine the exact 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  .

Consider the curve with equation  ,  defined for all values of  .

Find an expression for .

Hence determine the values of for which the curve is

(i)     concave up

(ii)    concave down.

Your answers should be given as exact values.

Use your answer to part (b) to show that the curve 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   ,  giving your answers as exact values.

An ice sculptor has created an abstract minimalist ice sculpture in the shape of a cylinder with radius  and height .  The sculpture is of solid ice throughout.

After a power cut that shuts off the sculptor’s freezer, the sculpture begins melting such that the volume of ice is decreasing at a constant rate of   per hour.

Assuming that while it melts the sculpture remains at all times in the shape of a cylinder which is mathematically similar to the original cylinder, find the rate at which the sculpture’s surface area is changing at the point when its radius is . 

A hemispherical bowl is supported with its curved surface on the bottom,  such that the plane defined by the open top of the bowl is at all times horizontal.  The bowl contains liquid, with the volume of liquid in the bowl being given by the formula

where  is the radius of the bowl and  is the depth of liquid (i.e., the height between the bottom of the bowl and the surface level of the liquid). 

The bowl is leaking liquid through a small hole in its bottom at a rate directly proportional to the depth of liquid.

Show that the rate of change of the depth of liquid in the bowl is

where  is a positive constant.

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   .

Show that the equation of the tangent to the graph of at point A may be expressed in the form 

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 curve with equation  defined for all  , where    is a positive integer.

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

In terms of , state in general how many (i) turning points and (ii) points of inflection the curve 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

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.  In your work you may use without proof the fact that for odd integers with 

A small conical flask, in the shape of a right cone stood on its flat base, is being filled with perfume via a small hole at its vertex.  The cone has a height of 6 cm and a radius of 2 cm. 

Perfume is being poured into the flask at a constant rate of 0.3 cm³s^-1.

Find the rate of change of the depth of the perfume in the flask at the instant when the flask is half full by volume.

A large block of ice is being prepared for use by a team of ice sculptors.  The block is in the shape of a cuboid with the ratio of its length to width to height being equal to  1 : 2 : 5. The block melts uniformly such that its surface area decreases at a constant rate, losing  of surface area every hour.  You may assume that as the block melts, its shape remains a cuboid with the dimensions in the same ratio to each other as in the original cuboid.

The block of ice is considered stable enough to be sculpted so long as the loss of volume due to melting does not exceed a rate  per hour. 

Find, in terms of , the volume of the largest block of ice that can be used for ice sculpting under such conditions.