Differentiation (OCR A Level Maths: Pure)

Prove, from first principles, that the derivative of -3x is -3.

Prove, from first principles, that the derivative of 2x² is 4x.

For each of the following, find  in terms of x:

Given that , , find .

For each of the following, find    in terms of x:

The function f is defined by .

Find .

Solve the equation .

A curve has the equation .

Find .

Find the coordinates of the point on the curve where the gradient is 2.

The function f is defined by

Find .

Given that the equation has exactly one real solution, find the value of c.

A curve is described by the equation .  

Make y the subject of the equation.

Hence find .

Find the coordinates of the point on the curve where the gradient is -2.

The curve with equation has a gradient of -7 at the point , and a gradient of -3 at the point (1, 3).

By considering show that and .

Hence find the values of a and b.

By considering a point that you know to be on the curve, find the value of c.

Prove, from first principles, that the derivative of is 2ax, where a is a constant.

Prove, from first principles, that the derivative of is .

For each of the following, find    in terms of x:

Given that   , find .

For each of the following, find  in terms of :

The function f is defined by . Show that there are no solutions to the equation .

A curve has the equation .

Show that  , where a and b are rational numbers to be found.

Hence find the coordinates of the point on the curve where the gradient is 0.

A curve has the equation , where b is a constant. Given that there is only one point on the curve where the gradient is zero, determine the possible values of b.

A curve is described by the equation  .  

By rearranging the equation to make y the subject, find .

The curve with equation has a gradient of 8 at the point , and a gradient of -10 at the point . Find the values of and c.

Prove, from first principles, that the derivative of    is  .

Show that

Prove, from first principles, that the derivative of is  .

For each of the following, find    in terms of :

Given that , find .

For each of the following, find    in terms of x:

The function f is defined by. Determine the range of values for p for which the equation has at least one real solution.

A curve has the equation ,  . Find the coordinates of the point on the curve where the gradient is 0.

The function f is defined by  . Determine the relationship between the value of n and the number of real solutions to the equation .

A curve is described by the equation  .  Find .

The curve with equation passes through the point (-1, 4).  At the point (2, 7) the gradient of the curve is 7.  Find the values of a, b and c.