Implicit Differentiation (A Level only) (OCR A Level Maths: Pure)

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

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.

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

.

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.

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

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.