Further Differentiation (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

(i) Differentiate 4 cos x - 3 sin x

(ii) Use the product rule to differentiate e^x In x

(iii) Use the quotient rule to differentiate

(iv) Use the chain rule to differentiate cos(x² - 7x + 1)

A curve has the equation y = e ^-3x + Inx, x> 0.

Find the gradient of the normal to the curve at the point (1, e^-3), giving your answer correct to 3 decimal places.

Find for each of the following:

y = cos(x² - 3x + 7) + sin(e^x)

y = In (2x³)

Find the equation of the tangent to the curve y = at the point (-2, 1),

giving your answer in the form ax + by + c = 0, where a, b and c are integers.

Differentiate with respect to x, simplifying your answers as far as possible:

(4 cos x - 3 sin x)e ^3x-5

(x³ - 4x² + 7) In x

Differentiate with respect to x.

Show that if y = cosec 2x, then

= - 2 cosec 2x cot 2x

Hence find the gradient of the tangent to the curve y = cosec 2x at the point with coordinates

The diagram below shows part of the graph of y = f(x), where f(x) is the function

defined by

f(x) = (x² - 1) In (x+ 3), ×>-3

Points A, B and C are the three places where the graph intercepts the x-axis.

Find f'(x).

Show that the coordinates of point A are (-2, 0).

Find the equation of the tangent to the curve at point A.

Use an appropriate method to differentiate each of the following.

(i) sin 2x - e^7x

(ii) x² In x

(iii)

(iv) In(tan x)

A curve has the equation y = e^-3x + In x, x > 0

Show that the equation of the tangent to the curve at the point with x-coordinate 1 is

For y = In(axⁿ), where a > 0 is a real number and n ≥ 1 is an integer, show that

Find the gradient of the normal to the curve y = 5 cos(e^x - ) at the point with x-coordinate 0. Give your answer correct to 3 decimal places.

Differentiate with respect to x, simplifying your answers as far as possible:

(2 sin 3х - cos 3x)e^6-×

(x²-x)² In 5x

By writing у = as y = f(x)[g(x)]^-1 and then using the product and chain rules,
show that

Given that x = sec 7y,

Find in terms of y

Hence find in terms of x.

The diagram below shows part of the graph of y = f(x), where f(x) is the function defined by

Point A is a maximum point on the graph.

Show that the x-coordinate of A is a solution to the equation

Use an appropriate method to differentiate each of the following.

(i)

(ii) (x² + 2x - 8) cos(3- x)

(iii)

(iv)

A curve has the equation y = 3^x + 2^-x.
Show that the gradient of the normal to the curve at the point

Show that the derivative y = 4^-×4 is

Hence find the equation of the tangent to the curve at the point (1,), giving your answer in the form y = ax + b, where a and b are to be given as exact values.

Differentiate with respect to x, simplifying your answers where possible:

The diagram below shows the graph of y = f(x), where f(x) is the function defined by

The points A and B are maximum and minimum points, respectively.

Find the range of f(x), giving your answer correct to 3 decimal places.