Differentiation (Edexcel IGCSE Further Maths): Exam Questions

$y = \mathrm{e}^{x ( x^{2} } - 3 x )$

Show that $y-2\frac{dy}{dx}+\frac{d^{2}y}{dx^{2}}=2e^{x}$

$y = \mathrm{e}^{2 x} ( x^{2} - 5 x )$

Show that $2e^{2x}=\frac{d^{2}y}{dx^{2}}-4\frac{dy}{dx}+4y$

The curve $C$ has equation $y = \mathrm{e}^{3 x} ( 2 x - 1 )^{4}$

Using calculus, find the exact value of the gradient of the tangent to $C$ when $x=1$

$y=(\mathrm{sin}2x)\sqrt{3+2x}$

Show that $\frac{dy}{dx}=\frac{\mathrm{sin}2x+(A+Bx)\mathrm{cos}2x}{\sqrt{3+2x}}$ where $A$and $B$are integers to be found.

$y=\frac{2e^{3x+1}}{5x^{2}}$

Find $\frac{dy}{dx}$

Give your answer in the form $\frac{Ae^{3x+1}(Bx-A)}{Cx^{3}}$ where $A$, $B$ and $C$ are prime numbers to be found.

The value of $x$ increases by 2%

Use your answer to part (a) to find an estimate, in terms of $x$ , for the percentage change in $y$ Give your answer in the form $( P x - Q )$ where $P$ and $Q$ are integers.

$y=e^{2x}\mathrm{cos}2x$

Show that

$\frac{dy}{dx}=2y-2e^{2x}\mathrm{sin}2x$

Hence show that

$\frac{d^{2}y}{dx^{2}}=4\frac{dy}{dx}-8y$

Figure 3 shows part of the curve $C$ with equation $y=4-e^{2x}$ The curve $C$ crosses the $y$-axis at the point $A$and the $x$-axis at the point $B$.

(i) Write down the $y$ coordinate of point $A$.

(ii) Show that the $x$ coordinate of $B$ is $x=\mathrm{ln}2$

The line $l$ is the normal to $C$ at the point $B$.

Find an equation for $l$ , giving your answer in the form $y=mx+c$

The finite region $R$ is bounded by $C$, $l$ and the $y$-axis.

Using calculus, find the area of $R$. Give your answer to one decimal place.