Differentiation (Cambridge (CIE) IGCSE Maths: Extended): Exam Questions

$\mathrm{f}\left(x\right)=x(x+2)(x–3)$

On the diagram, sketch the graph of $y=\mathrm{f}\left(x\right)\mathrm{for}–3⩽x⩽4$ .
Show the values of the intersections with the axes. 

Expand and simplify.

$x(x+2)(x–3)$

$A$ is the point $(1,–6)$.
The tangent to the graph of $y=\mathrm{f}\left(x\right)$ at  $A$ meets the $y$-axis at $B$.

Find the coordinates of $B$.

$y=x^4-4x^3$

Find the two stationary points on the graph of $y=x^4-4x^3.$  

( ..................... , ..................... )
( ..................... , ..................... )

A curve has equation $y=x^3-3x+4$.

Work out the coordinates of the two stationary points.

( .................... , ....................)
( .................... , ....................)

Determine whether each stationary point is a maximum or a minimum. Give reasons for your answers.

$y=x^3–6x^2–15x$.

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.

$\frac{\mathrm{d}y}{\mathrm{d}x}=$....................................

The curve with equation $y=x^3–6x^2–15x$ has two stationary points.

Work out the coordinates of these two stationary points.

Expand and simplify $\left(x-2\right)\left(x-4\right)\left(x+2\right)$

Given that $\mathrm{f}\left(x\right)=\left(x-2\right)\left(x-4\right)\left(x+2\right)$, find the gradient of the tangent to the curve $y=\mathrm{f}\left(x\right)$ at $x=1$.

 The diagram shows a sketch of the curve  $y=x^2+3x-4$.

i) Differentiate  $y=x^2+3x-4$

   [2]

ii) Find the equation of the tangent to the curve at the point (2, 6). 

 [3]

$y=x^p+2x^q$

$\frac{dy}{dx}=11x^{10}+10x^4$, where $\frac{dy}{dx}$ is the derived function.  

Find the value of $p$ and the value of $q$.  

$p=$................................................

$q=$................................................

A curve, C, has equation $y=2x^2+8k^{2}x-3$ where k is a constant.

Show that when k = 0, the turning point on C has coordinates (0, -3).

Show that when $k\ne 0$, the turning point on C must have a negative x-coordinate.

When $k\ne 0$ determine whether or not the $y$-coordinate of the turning point is negative.