Differentiating powers of x

The equation of a curve is y = x⁵ + px⁴ where p is a constant. When x = – 1 the gradient of the curve is – 3

Work out the value of p.

For a curve y = f(x)

Show that the curve has a turning point where x = 3

Use to work out the nature of the turning point. You must show your working.

Here is a sketch of the curve with equation y = 4 + 2x – x² The tangent to the curve at the turning point P is shown. The normal to the curve at Q meets this tangent at R.

Work out the x-coordinate of R.

The equation of a curve is y = x⁴ – 3kx² where k is a constant. When x = 2 the rate of change of y with respect to x is 23

Work out the value of k.

Give your answer in the form a + bxⁿ where a, b and n are integers.

The equation of a curve is y = x⁶ + 4x² – 7

Work out the equation of the tangent to the curve at the point (1, – 2) Give your answer in the form y = mx + c

Work out the positive value of x for which = 55

The curve y = ax³ + bx² + 7 has a stationary point at (–2, 11)

Work out the values of a and b.

y = 0.5x⁴

Work out the value of x for which the rate of change of y with respect to x is 6.75

The function h is given by h(x) = ax(3x² – 2) + 5x where a is a positive constant. h is an increasing function for all values of x.

Work out the possible values of a. Give your answer as an inequality.

The curve y = x⁴ – 18x² has three stationary points.

Work out the coordinates of the three stationary points and determine their nature. You must show your working.

A curve has equation y = x³ – 5x² At two points on the curve, the rate of change of y with respect to x is 4

Work out an equation, in terms of x, to represent this information. Give your answer in the form ax² + bx + c = 0 where a, b and c are integers.

Hence, work out the two possible values of x. Give your answers to 3 significant figures.

at the point where x = – 1 You must show your working.

y = x(2x⁴ – 7x³)

Work out an expression for the rate of change of y with respect to x.

P is the point on the curve y = ax³ + 10x² where x = 2 The gradient of the normal to the curve at P is –

Work out the value of a.