Further Pure Math (Differentiation)

 1. (2011/june/paper01/q3)

Given that $y=\mathrm{e}^{2 x} \sin 3 x$

(a) find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$

(b) show that $\dfrac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2 \dfrac{\mathrm{d} y}{\mathrm{~d} x}-9 y+6 \mathrm{e}^{2 x} \cos 3 x$

2. (2012/jan/paper01/q5)

Differentiate with respect to $x$

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

(b) $y=\left(x^{3}+2 x^{2}+3\right)^{5}$

3. (2012/june/paper02/q2)

Given that $x=t^{3}+4$ and $y=1-t+5 t^{2}$

(a) find

(i) $\dfrac{\mathrm{d} x}{\mathrm{~d} t}$

(ii) $\dfrac{\mathrm{d} y}{\mathrm{~d} t}$ (2)

(b) Find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$ in terms of $t$

4. $(2012 /$ june $/$ paper02/q4)

Differentiate with respect to $x$

(a) $\dfrac{1}{x^{2}}$

(b) $\dfrac{1}{(2 x+1)^{2}}$

(c) $\dfrac{1}{1-\cos ^{2} x}$

5. $(2013 / \mathrm{jan} /$ paper02/q4)

Differentiate with respect to $x$

(a) $3 x \sin 5 x$

(b) $\dfrac{e^{2 x}}{4-3 x^{2}}$

6. (2014/jan/paper01/q3)

Differentiate with respect to $x$

(a) $\mathrm{e}^{3 x}(5 x-7)^{2}$

(b) $\dfrac{\cos 2 x}{x+9}$

7. $(2015 /$ june $/$ paper01/q2)

Given that $y=4 x^{2} \mathrm{e}^{2 x}$

(a) find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$

(b) hence show that $x \dfrac{\mathrm{d} y}{\mathrm{~d} x}=2 y(1+x)$

8. $(2016 / \mathrm{jan} / \mathrm{paper} 01 / \mathrm{q} 1)$

$$\mathrm{f}(x)=3 x^{3}+2 \sin x-\dfrac{4}{x^{2}} \text { where } x \neq$$

(a) Find $\mathrm{f}^{\prime}(x)$

(b) Find $\int \mathrm{f}(x) \mathrm{d} x$

9. $(2016 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 4)$

Given that $y=\mathrm{e}^{2 x} \sqrt{x+1}$

show that $\dfrac{\mathrm{d} y}{\mathrm{~d} x}=\dfrac{\mathrm{e}^{2 x}(4 x+5)}{2 \sqrt{x+1}}$

10. (2016/june/paper02/q4)

Differentiate with respect to $x$

$$\mathrm{e}^{2 x} \cos 3 x$$

11. ( $2018 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 5)$

Given that $y=2 \mathrm{e}^{*}\left(3 x^{2}-6\right)$

show that $\dfrac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}-2 \dfrac{\mathrm{d} y}{\mathrm{~d} x}+y=12 \mathrm{e}^{x}$

12. $(2018 /$ june $/$ paper02 $/ \mathrm{q} 2$ )

Differentiate with respect to $x$

(a) $e^{3 x} \cos 2 x$

(b) $\dfrac{2 \mathrm{e}^{t}}{\left(2 x^{2}-1\right)}$ (3)

13. $(2019 /$ june $/$ paper02 $/ \mathrm{q} 6$ )

(a) Given that $y=(4 x-3) \mathrm{e}^{2 *}$

(i) find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$

(ii) show that $(4 x-3) \dfrac{\mathrm{d} y}{\mathrm{~d} x}=(8 x-2) y$

(b) Differentiate $\dfrac{\sin 5 x}{(x-3)^{2}}$ with respect to $x$

Answer

1.(a) $\dfrac{d y}{d x}=2 e^{2 x} \sin 3 x+3 e^{2 x} \cos 3 x$ (b) Show

2.(a)$\dfrac{d y}{d x}=x^{2} e^{x}+2 x e^{x} \quad$ (b) $\dfrac{d y}{d x}=5\left(x^{3}+2 x^{2}+3\right)^{4}\left(3 x^{2}+4 x\right)$

3.(a)(i) $\dfrac{d x}{d t}=3 t^{2}$ (ii) $\dfrac{d y}{d t}=-1+10 t$ (b) $\dfrac{d y}{d x}=\dfrac{10 t-1}{3 t^{2}}$

4.(a) $\dfrac{2}{x^{3}}$ (b) $\dfrac{4}{(2 x+1)^{3}}$ (c) $\dfrac{-2 \cos x}{\sin ^{3} x}$

5.(a) $3 \sin 5 x+15 x \cos 5 x$ (b) $\dfrac{2 e^{2 x}\left(4-3 x^{2}\right)-e^{2 x}(-6 x)}{\left(4-3 x^{2}\right)^{2}}$

6.(a) $3 e^{3 x}(5 x-7)^{2}+10 e^{3 x}(5 x-7)$ (b) $\dfrac{-2 \sin (2 x)(x+9)-\cos 2 x}{(x+9)^{2}}$

7. (a) $\dfrac{d y}{d x}=8 x e^{2 x}+8 x^{2} e^{2 x}$(b) Show

8. $(a) f^{\prime}(x)=9 x^{2}+2 \cos x+8 x^{-3}$ (b) $\dfrac{3 x^{4}}{4}-2 \cos x-\dfrac{4 x^{-1}}{-1}+C$

9. show

10. $2 e^{2 x} \cos 3 x-3 e^{2 x} \sin 3 x$

11. Show

12. $\begin{aligned}

13. (a) (i) $\dfrac{d y}{d x}=4 e^{2 x}+2(4 x-3) e^{2 x}$ (ii) Show (b) $\dfrac{d y}{d x}=-2(x-3)^{-3} \sin 5 x+5(x-3)^{-2} \cos 5 x$