$$\begin{aligned} & \begin{aligned} \text { Let } \quad y & =\left(a x^2+\cot x\right)(p+q \cos x) \\ \therefore \quad \frac{d y}{d x} & =\left(a x^2+\cot x\right) \frac{d}{d x}(p+q \cos x)+(p+q \cos x) \frac{d}{d x}\left(a x^2+\cot x\right) \quad \text { [by product rule] } \\ & =\left(a x^2+\cot x\right)(-q \sin x)+(p+q \cos x)\left(2 a x-\operatorname{cosec}^2 x\right) \\ & =-q \sin x\left(a x^2+\cot x\right)+(p+q \cos x)\left(2 a x-\operatorname{cosec}^2 x\right) \end{aligned} \end{aligned}$$

$$\begin{aligned} &\text { Let } \quad y=\frac{a+b \sin x}{c+d \cos x}\\ &\begin{aligned} \therefore \quad \frac{d y}{d x} & =\frac{(c+d \cos x) \frac{d}{d x}(a+b \sin x)-(a+b \sin x) \frac{d}{d x}(c+d \cos x)}{(c+d \cos x)^2} \text { [by quotinet rule] } \\ & =\frac{(c+d \cos x)(b \cos x)-(a+b \sin x)(-d \sin x)}{(c+d \cos x)^2} \\ & =\frac{b c \cos x+b d \cos ^2 x+a d \sin x+b d \sin ^2 x}{(c+d \cos x)^2} \\ & =\frac{b c \cos x+a d \sin x+b d\left(\cos ^2 x+\sin ^2 x\right)}{(c+d \cos x)^2} \\ & =\frac{b c \cos x+a d \sin x+b d}{(c+d \cos x)^2} \end{aligned} \end{aligned}$$

$$\begin{aligned} &\text { Let } \quad y=(\sin x+\cos x)^2\\ &\begin{aligned} \therefore \quad \frac{d y}{d x} & =2(\sin x+\cos x)(\cos x-\sin x)\\ & =2\left(\cos ^2 x-\sin ^2 x\right) \quad \text{[by chain rule]}\\ & =2 \cos 2 x\quad [\because \cos^2x=\cos^2x-\sin^2x] \end{aligned} \end{aligned}$$

$$\begin{aligned} \text{Let}\quad y & =(2 x-7)^2(3 x+5)^3 \\ \frac{d y}{d x} & =(2 x-7)^2 \frac{d}{d x}(3 x+5)^3+(3 x+5)^3 \frac{d}{d x}(2 x-7)^2 \quad \text{[by product rule]}\\ & =(2 x-7)^2(3)(3 x+5)^2(3)+(3 x+5)^3 2(2 x-7)(2) \quad \text{[by chain rule]}\\ & =9(2 x-7)^2(3 x+5)^2+4(3 x+5)^3(2 x-7) \\ & =(2 x-7)(3 x+5)^2[9(2 x-7)+4(3 x+5)] \\ & =(2 x-7)(3 x+5)^2(18 x-63+12 x+20) \\ & =(2 x-7)(3 x+5)^2(30 x-43) \end{aligned}$$

Let $$ \begin{aligned} y & =x^2 \sin x+\cos 2 x \\ \frac{d y}{d x} & =\frac{d}{d x}\left(x^2 \sin x\right)+\frac{d}{d x} \cos 2 x \\ & =x^2 \cdot \cos x+\sin x 2 x+(-\sin 2 x) \cdot 2 \quad \text{[by product rule] }\\ & =x^2 \cos x+2 x \sin x-2 \sin 2 x\quad \text{[by chain urle]} \end{aligned}$$