Sec 4 A-Math Differentiation topical questions

Differentiate with respect to x: 6x³ − 2/x² + 4√x.

Find the equation of the tangent to the curve y = x² − 4x at the point where x = 5.

Find the equation of the normal to the curve y = x³ at the point where x = 2, giving your answer in the form ax + by = c.

Show that if y = x√(4x − 1), then dy/dx = (6x − 1)/√(4x − 1).

Find the coordinates and nature of the stationary points of y = x³ − 12x + 2.

For the curve y = x⁴ − 4x³, find the stationary values and classify each, using the first derivative test where the second derivative test fails.

The volume of a spherical balloon is V = (4/3)πr³ and air enters at a constant rate of 8 cm³/s. Find the rate at which the radius is growing when r = 2 cm, giving your answer correct to 3 significant figures.

A closed cylinder has volume 54π cm³. (a) Show that its total surface area is A = 2πr² + 108π/r. (b) Find the radius and height that minimise A, justifying that the value is a minimum, and state the minimum surface area.

Find the exact coordinates and nature of the stationary point of y = xe^(−x), and hence state the maximum value of y.

The curve y = 4 sin x − 2 sin 2x is defined for 0 ≤ x ≤ π. (a) Show that dy/dx = 4 cos x − 4 cos 2x. (b) Show that the stationary points occur where cos x = 1 or cos x = −1/2. (c) Find the maximum value of y for 0 ≤ x ≤ π, giving your answer in exact form.