Differentiate the following w.r.t x

(i) $y=\sqrt{sin x^3}$

$\frac{dy}{dx}=\frac{1}{sin x^3} \frac{d}{dx}(sin x^3)$

$=\frac{1}{2\sqrt{sin x^3}} cos x^3\frac{d}{dx} x^3$

$=\frac{cos x^3}{sin x^3}3x^2$

$=\frac{3x^2 cos x^3}{sin x^3}$

(ii) $y = \log [\cos (x^{5})]$ $y=log[cos(x^5)]$

$\frac{d y}{d x} = \frac{d}{d x} (\log [\cos (x^{5})])$ $\frac{dy}{dx}=\frac{d}{dx}(log[cos(x^5)])$

$= \frac{1}{\cos (x^{5})} \frac{d}{d x} [\cos (x^{5})]$ $=\frac{1}{cos(x^5)}\frac{d}{dx}[cos(x^5)]$

$= \frac{1}{\cos (x^{5})} - {\sin x}^{5} \frac{d}{d x} x^{5}$ $=\frac{1}{cos(x^5)}-sinx^5\frac{d}{dx} x^5$

$= \frac{- {\sin x}^{5}}{{\cos x}^{5}} 5 x^{3}$ $=\frac{-sin x^5}{cos x^5}5x^3$

$= \frac{- 5 x^{3}}{{\cos x}^{5}}$ $=\frac{-5x^3}{cos x^5}$

(iii) $y = \cot^{2} (x^{3})$ $y=cot^{2}(x^3)$

$\frac{d y}{d x} = 2 \cot (x^{3}) \frac{d}{d x} [\cot (x^{3})]$ $\frac{dy}{dx}=2cot(x^3)\frac{d}{dx}[cot(x^3)]$

$= 2 \cot (x^{3}) - \cos e c x^{3} \frac{d}{d x} x^{3}$ $=2cot(x^3)-cosec x^3 \frac{d}{dx} x^3$

$= - 2 \cot (x^{3}) \cos e c^{2} (x^{3}) 3 x^{2}$ $=-2cot(x^3)cosec^2(x^3) 3x^2$

$= - 6 x^{2} \cot (x^{3}) \cos e c^{2} (x^{3})$ $=-6x^2cot(x^3)cosec^2(x^3)$

(iv) $y = \log [\cos (x^{5})]$ $y=log[cos(x^5)]$

$\frac{d y}{d x} = \frac{1}{\cos (x^{5})} \times \frac{d}{d x} \cos (x^{5})$ $\frac{dy}{dx}=\frac{1}{cos(x^5)}\times\frac{d}{dx}cos(x^5)$

$= \frac{1}{\cos (x^{5})} \times - \sin (x^{5}) \frac{d}{d x} x^{5}$ $=\frac{1}{cos(x^5)}\times -sin(x^5)\frac{d}{dx}x^5$

$= \frac{- \sin (x^{5})}{\cos (x^{5})} 5 x^{4}$ $= \frac{ -sin(x^5)}{cos(x^5)}5x^4$

$= \frac{- 5 x^{4} \sin (x^{5})}{\cos (x^{5})}$ $= \frac{ -5x^4sin(x^5)}{cos(x^5)}$

(v) $y = {(x^{3} + 2 x - 3)}^{4} {(x + \cos x)}^{3}$ $y=(x^3+2x-3)^4(x+cos x)^3$

$\frac{d y}{d x} = 4 {(x^{3} + 2 x - 3)}^{3} \frac{d}{d x} x^{3} + 2 x - 3 \times 3 {(x + \cos x)}^{2} \frac{d}{d x} x + \cos x$ $\frac{dy}{dx}=4(x^3+2x-3)^3\frac{d}{dx}x^3+2x-3 \times 3(x+cos x)^2\frac{d}{dx}x+cos x$

$= 4 {(x^{3} + 2 x - 3)}^{3} (3 x^{2} + 2) \times 3 {(x + \cos x)}^{2} (1 - \sin x)$ $=4(x^3+2x-3)^3\left(3x^2+2\right) \times 3(x+cos x)^2 (1-sin x)$

(vi) $y = {(1 + \cos^{2} x)}^{4} \times \sqrt{x + \sqrt{\tan x}}$ $y=(1+cos^2 x)^4 \times \sqrt{x+\sqrt{tan x}}$

$\frac{d y}{d x} = \frac{d}{d x} {(1 + \cos^{2} x)}^{4}$ $\frac{dy}{dx}=\frac{d}{dx}(1+cos^2 x)^4$ $\times \sqrt{x + \sqrt{\tan x}}$ $\times \sqrt{x+\sqrt{tan x}}$

$= {(1 + \cos^{2} x)}^{4}$ $=(1+cos^2 x)^4$ $\frac{d}{d x} \sqrt{x + \sqrt{\tan x}}$ $\frac{d}{dx}\sqrt{x+\sqrt{tan x}}$ $+ \sqrt{x + \sqrt{\tan x}}$ $+ \sqrt{x+\sqrt{tan x}}$ $\frac{d}{d x} {(1 + \cos^{2} x)}^{4}$ $\frac{d}{dx}(1+cos^2 x)^4$

$= {(1 + \cos^{2} x)}^{4}$ $=(1+cos^2 x)^4$ $\frac{1}{2 \sqrt{x + \sqrt{\tan x}}}$ $\frac{1}{2\sqrt{x+\sqrt{tan x}}}$ $\frac{d}{d x} (x + \sqrt{\tan x})$ $\frac{d}{dx}(x+\sqrt{tan x})$ $+ \sqrt{x + \sqrt{\tan x}}$ $+ \sqrt{x+\sqrt{tan x}}$ $\times$ $\times$ $4 {(1 + \cos^{2} x)}^{3}$ $4 (1+cos^2 x)^3$ $\frac{d}{d x} (1 + \cos^{2} x)$ $\frac{d}{dx}(1+cos^2 x)$

$= {(1 + \cos^{2} x)}^{4}$ $=(1+cos^2 x)^4$ $\frac{1}{2 \sqrt{x + \sqrt{\tan x}}}$ $\frac{1}{2\sqrt{x+\sqrt{tan x}}}$ $(1 + \frac{1}{2 \sqrt{\tan x}})$ $\left(1 + \frac{1}{2\sqrt{tan x}}\right)$ $\frac{d}{d x} \tan x$ $\frac{d}{dx}tanx$ $+ \sqrt{x + \sqrt{\tan x}}$ $+ \sqrt{x+\sqrt{tan x}}$ $\times$ $\times$ $4 {(1 + \cos^{2} x)}^{3} \times$ $4 (1+cos^2 x)^3\times$ $\frac{d}{d x} (1 + \cos^{2} x)$ $\frac{d}{dx}(1+cos^2 x)$

$= {(1 + \cos^{2} x)}^{4}$ $=(1+cos^2 x)^4$ $\frac{1}{2 \sqrt{x + \sqrt{\tan x}}}$ $\frac{1}{2\sqrt{x+\sqrt{tan x}}$ $(1 + \frac{1}{2 \sqrt{\tan x}})$ $\left(1 + \frac{1}{2\sqrt{tan x}}\right)$ $\sec^{2} x \sqrt{x + \sqrt{\tan x}}$ $sec^2x \sqrt{x+\sqrt{tan x}}$ $\times$ $\times$ $4 {(1 + \cos^{2} x)}^{3}$ $4 (1+cos^2 x)^3$ $\times$ $\times$ $2 \cos x$ $2cos x$ $\frac{d}{d x} \cos x$ $\frac{d}{dx}cos x$

$= {(1 + \cos^{2} x)}^{4}$ $=(1+cos^2 x)^4$ $\frac{1}{2 \sqrt{x + \sqrt{\tan x}}}$ $\frac{1}{2\sqrt{x+\sqrt{tan x}}$ $\times$ $\times$ $1 + \frac{1}{2 \sqrt{\tan x}}$ $1 + \frac{1}{2\sqrt{tan x}$ $\sec^{2} x$ $sec^2x$ $\sqrt{x + \sqrt{\tan x}}$ $\sqrt{x+\sqrt{tan x}}$ $\times 4$ $\times 4$ ${(1 + \cos^{2} x)}^{3}$ $(1+cos^2 x)^3$ $\times$ $\times$ $- 2 \cos x \sin x$ $-2cos x sinx$

Differentiate the following w.r.t x

Differentiate the following w.r.t x