Differentiation (Edexcel IGCSE Further Pure Maths): Exam Questions

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

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

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

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

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

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

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

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

Give your answer in the form $\frac{A{e}^{3x+1}(Bx-A)}{C{x}^3}$ 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 $\left(Px-Q\right)$ where $P$ and $Q$ are integers.

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

Show that

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

Hence show that

$\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=4\frac{\mathrm{d}y}{\mathrm{d}x}-8y$

Figure 3 shows part of the curve $C$ with equation $y=4-{\mathrm{e}}^{2x}$
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 $x=\mathrm{In}2$

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.