Implicit Differentiation (Edexcel A Level Maths: Pure): Exam Questions

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

A curve has equation

Find in terms of and .

The point lies on the curve.

Find the equation of the normal to the curve at , giving your answer in the form , where , and are integers to be found.

Find an expression for in terms of x and y for the following

(i)

(ii)

Find the gradient of the curve with equation

at the point with coordinates (2 , 3).

The curve C has equation

Show that C intersects the x-axis at the points (-5 , 0) and (5 , 0).

Find an expression for in terms of x and y.

Hence find the gradients of C at the two points where C intercepts the x-axis.

Show that if, then 

Show that the point (0 , π) lies on the curve with equation

Find an expression for .

Find the gradient at the point (0, π).

Hence find an equation of the tangent to the curve at the point (0 ,π).

The curve C is defined by the equation .

The point P(1 , 1) lies on C.

Show that

Find the gradient of the tangent to C at point P, and hence find the gradient of the normal to C at point P.

Find the equation of the normal to C at point P, giving your answer in the form , where a, b and c are integers to be found.

Find an expression for in terms of x and y given that

Show that when.

Hence, or otherwise, find the coordinates of the stationary points.

The curve C is given by the equation 

Find the coordinates of the points where C  intersects the coordinate axes.

Find an expression for

Show that the tangents to C, at the points where it meets the coordinate axes, have equations

and  

The tangents meet at point Q.
Find the distance OQ, where O is the origin.

Use implicit differentiation to show that

where a is a constant.

Figure 8 shows a sketch of the curve with equation .

Find, by firstly taking logarithms, the coordinate of the turning point of .

(Solutions based entirely on graphical or numerical methods are not acceptable.)

The curve has equation

Show that

Prove that has a point of inflection at .

Find an expression for in terms of x  and y for the following

(i)

(ii) .

Given that

find the positive value of x when .

Hence, or otherwise, find the value of the gradient of

at the point where and x is positive.

Show that if , then

The curve C has equation , Line L has equation .

Show that the two points where C intersects L have equal gradients.

What else can you deduce about the two points where C and L intercept?

Verify that the point (-1 , 0) lies on the curve with equation

and find the equation of the tangent to the curve at the point (-1 , 0).
Give your answer in the form , where a, b and c are integers to be found.

Show that the derivative function of the curve given by 

is given by

Find the equation of the normal to the curve given in part (a) at the point where , giving your answer in the form where and c are integers to be found.

Show that the stationary points on the curve with equation

occur when , and find the exact y-coordinates of the stationary points.

Verify that the point A(1 , 1) lies on the curve with equation

The tangent at point A intercepts the x-axis at point B and the y-axis at point C.

Find the area of the triangle OBC.

Show that

where a and k are constants.

The curve has equation

where and are constants.

Show that

where , and are integers to be found.

Given that

• the point lies on

• the normal to at has equation

find the value of and the value of .

Figure 4 shows a sketch of the curve with equation .

Show that .

The curve is used to model the shape of a cycle track with both and measured in km.

The points and represent points that are furthest west and furthest east of the origin , as shown in Figure 4.

Using part (a), find the exact coordinates of the point .

Explain briefly how to find the coordinates of the point that is furthest north of the origin . (You do not need to carry out this calculation).

Find an expression for in terms of x and y for the following

(i)

(ii)

Find the gradient at the point where and is an integer on the curve with equation 

Show that if , then

An ellipse has equation

Find an expression for and hence show that the gradient of the ellipse at any point where it meets a line of the form is independent of x and y.

The curve C is described by the equation

Show that the tangents of the two points on C wheremeet at the point .

The curve C is described by the equation

Show that the normal to C at the point where is parallel to the normal to C at the point where

Find the distance between the y-axis intercepts of these two normals.

Find the stationary points and determine their nature for the curve with equation 

The curve C is defined by

Points A and B have coordinates and respectively.

The tangents to C at points A and B intersect at the point P.
The tangent to C at point A intersects the x-axis at point Q.
The tangent to C at point B intersects the x-axis at point R.

Find the area of triangle PQR.

Show that

where a and k are constants.