RD Sharma solutions for Mathematics [English] Class 12 chapter 11 - Differentiation [Latest edition]

Differentiate the following functions from first principles e^−x.

Differentiate the following functions from first principles e^3x.

Differentiate the following functions from first principles e^ax+b.

Differentiate the following functions from first principles e^cos x.

Differentiate the following functions from first principles  $$e^{\sqrt{2x}}$$.

Differentiate the following functions from first principles log cos x ?

​Differentiate the following function from first principles $$e^{\sqrt{\cot x}}$$ .

Differentiate the following functions from first principles x²e^x ?

Differentiate the following functions from first principles log cosec x ?

Differentiate the following functions from first principles sin⁻¹ (2x + 3) ?

Differentiate sin (3x + 5) ?

Differentiate tan² x ?

Differentiate tan (x° + 45°) ?

Differentiate sin (log x) ?

Differentiate $$e^{\sin }\sqrt{x}$$ ?

Differentiate e^tan x ?

Differentiate sin² (2x + 1) ?

Differentiate log₇ (2x − 3) ?

Differentiate tan 5x° ?

Differentiate ${\displaystyle 2^{x^3}}$ ?

Differentiate $$3^{e^x}$$ ?

Differentiate log_x 3 ?

Differentiate $$3^{x^2+2x}$$ ?

Differentiate $$\sqrt{\frac{a^2-x^2}{a^2+x^2}}$$ ?

Differentiate $$3^{x\log x}$$ ?

Differentiate $$\sqrt{\frac{1+\sin x}{1-\sin x}}$$ ?

Differentiate $$\sqrt{\frac{1-x^2}{1+x^2}}$$ ?

Differentiate (log sin x)² ?

Differentiate $$\sqrt{\frac{1+x}{1-x}}$$ ?

Differentiate $$\sin \left(\frac{1+x^2}{1-x^2}\right)$$ ?

Differentiate $$e^{3x}\cos 2x$$ ?

Differentiate sin(log sin x) ?

Differentiate $$e^{\tan 3x}$$ ?

Differentiate $$e^{\sqrt{\cot x}}$$ ?

Differentiate $$\log \left(\frac{\sin x}{1+\cos x}\right)$$ ?

Differentiate $$\log \sqrt{\frac{1-\cos x}{1+\cos x}}$$ ?

Differentiate $$\tan \left(e^{\sin x}\right)$$ ?

Differentiate $$\log (x+\sqrt{x^2+1})$$ ?

Differentiate $$\frac{e^x\log x}{x^2}$$ ? 

Differentiate $$\log (cosecx-\cot x)$$ ?

Differentiate $$\frac{e^{2x}+e^{-2x}}{e^{2x}-e^{-2x}}$$ ?

Differentiate $$\log \left(\frac{x^2+x+1}{x^2-x+1}\right)$$ ?

Differentiate $${\tan }^{-1}\left(e^x\right)$$ ?

Differentiate $$e^{{\sin }^{-1}2x}$$ ?

Differentiate $$\sin (2{\sin }^{-1}x)$$ ?

Differentiate $$e^{{\tan }^{-1}}\sqrt{x}$$ ?

Differentiate $$\sqrt{{\tan }^{-1}\left(\frac{x}{2}\right)}$$ ?

Differentiate $$\log ({\tan }^{-1}x)$$? 

Differentiate $$\frac{2^x\cos x}{{(x^2+3)}^2}$$ ?

Differentiate $$x\sin 2x+5^x+k^k+{({\tan }^{2}x)}^3$$ ?

Differentiate $$\log (3x+2)-x^2\log (2x-1)$$ ?

Differentiate $$\frac{3x^2\sin x}{\sqrt{7-x^2}}$$ ?

Differentiate $${\sin }^2\{\log (2x+3)\}$$ ?

Differentiate  $$e^x\log \sin 2x$$ ?

Differentiate $$\frac{\sqrt{x^2+1}+\sqrt{x^2-1}}{\sqrt{x^2+1}-\sqrt{x^2-1}}$$ ?

Differentiate ${\displaystyle \log [x+2+\sqrt{x^2+4x+1}]}$

Differentiate $${({\sin }^{-1}{x}^4)}^4$$ ?

Differentiate $${\sin }^{-1}\left(\frac{x}{\sqrt{x^2+a^2}}\right)$$ ?

Differentiate $$\frac{e^x\sin x}{{(x^2+2)}^3}$$ ?

Differentiate $$3e^{-3x}\log (1+x)$$ ?

Differentiate $$\frac{x^2+2}{\sqrt{\cos x}}$$ ?

Differentiate $$\frac{x^2(1-x^2)}{\cos 2x}$$ ?

$$\log \{\cot (\frac{\pi }{4}+\frac{x}{2})\}$$ ?

Differentiate $$e^{ax}\sec x\tan 2x$$ ?

Differentiate $$\log (\cos x^2)$$ ?

Differentiate $$\cos {(\log x)}^2$$ ?

Differentiate $$\log \sqrt{\frac{x-1}{x+1}}$$ ?

If $$y=\log \{\sqrt{x-1}-\sqrt{x+1}\}$$ ,show that $$\frac{dy}{dx}=\frac{-1}{2\sqrt{x^2-1}}$$ ?

 If $$y=\sqrt{x+1}+\sqrt{x-1}$$ , prove that $$\sqrt{x^2-1}\frac{dy}{dx}=\frac{1}{2}y$$ ?

If $$y=\frac{x}{x+2}$$  , prove tha $$x\frac{dy}{dx}=(1-y)y$$ ? 

If $$y=\log (\sqrt{x}+\frac{1}{\sqrt{x}})$$prove that $$\frac{dy}{dx}=\frac{x-1}{2x(x+1)}$$ ?

If  $$y=\log \sqrt{\frac{1+\tan x}{1-\tan x}}$$  prove that $$\frac{dy}{dx}=\sec 2x$$ ?

If $$y=\sqrt{x}+\frac{1}{\sqrt{x}}$$, prove that  $$2x\frac{dy}{dx}=\sqrt{x}-\frac{1}{\sqrt{x}}$$ ?

If $$y=\frac{x{\sin }^{-1}x}{\sqrt{1-x^2}}$$ ,  prove that $$(1-x^2)\frac{dy}{dx}=x+\frac{y}{x}$$ ?

If $$y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$ .prove that $$\frac{dy}{dx}=1-y^2$$ ?

If  $$y=(x-1)\log (x-1)-(x+1)\log (x+1)$$ , prove that $$\frac{dy}{dc}=\log \left(\frac{x-1}{1+x}\right)$$ ?

If $$y=e^x\cos x$$ ,prove that $$\frac{dy}{dx}=\sqrt{2}{e}^x\cdot \cos (x+\frac{\pi }{4})$$ ?

If $$y=\frac{1}{2}\log \left(\frac{1-\cos 2x}{1+\cos 2x}\right)$$ , prove that $$\frac{dy}{dx}=2\text{cosec }2x$$ ?

If $$y=x{\sin }^{-1}x+\sqrt{1-x^2}$$ ,prove that $$\frac{dy}{dx}={\sin }^{-1}x$$ ?

If $$y=\sqrt{x^2+a^2}$$ prove that  $$y\frac{dy}{dx}-x=0$$ ?

If $$y=e^x+e^{-x}$$ prove that  $$\frac{dy}{dx}=\sqrt{y^2-4}$$ ?

If $$y=\sqrt{a^2-x^2}$$ prove that  $$y\frac{dy}{dx}+x=0$$ ?

If xy = 4, prove that $$x(\frac{dy}{dx}+y^2)=3y$$ ?

Prove that $$\frac{d}{dx}\{\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\sin }^{-1}\frac{x}{a}\}=\sqrt{a^2-x^2}$$ ?

Differentiate $${\cos }^{-1}\left\{2x\sqrt{1-x^2}\right\},\frac{1}{\sqrt{2}}<x<1$$ ?

Differentiate $${\cos }^{-1}\left\{\sqrt{\frac{1+x}{2}}\right\},-1<x<1$$ ?

Differentiate  $${\sin }^{-1}\left\{\sqrt{\frac{1-x}{2}}\right\},0<x<1$$  ?

Differentiate $${\sin }^{-1}\left\{\sqrt{1-x^2}\right\},0<x<1$$ ?

Differentiate $${\tan }^{-1}\left\{\frac{x}{\sqrt{a^2-x^2}}\right\},-a<x<a$$ ?

Differentiate $${\sin }^{-1}\left\{\frac{x}{\sqrt{x^2+a^2}}\right\}$$ ?

Differentiate $${\sin }^{-1}(2x^2-1),0<x<1$$  ?

Differentiate $${\sin }^{-1}(1-2x^2),0<x<1$$ ?

Differentiate $${\cos }^{-1}\left\{\frac{x}{\sqrt{x^2+a^2}}\right\}$$ ?

Differentiate $${\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\},-\frac{3\pi }{4}<x<\frac{\pi }{4}$$ ?

Differentiate $${\cos }^{-1}\left\{\frac{\cos x+\sin x}{\sqrt{2}}\right\},-\frac{\pi }{4}<x<\frac{\pi }{4}$$ ?

Differentiate $${\tan }^{-1}\left\{\frac{x}{1+\sqrt{1-x^2}}\right\},-1<x<1$$ ?

Differentiate $${\tan }^{-1}\left\{\frac{x}{a+\sqrt{a^2-x^2}}\right\},-a<x<a$$ ?

Differentiate $${\sin }^{-1}\left(\frac{x+\sqrt{1-x^2}}{\sqrt{2}}\right),-1<x<1$$ ?

Differentiate $${\cos }^{-1}\left(\frac{x+\sqrt{1-x^2}}{\sqrt{2}}\right),-1<x<1$$ ?

Differentiate $${\tan }^{-1}\left(\frac{4x}{1-4x^2}\right),-\frac{1}{2}<x<\frac{1}{2}$$ ?

Differentiate $${\tan }^{-1}\left(\frac{2^{x+1}}{1-4^x}\right),-\mathrm{\infty }<x<0$$ ?

Differentiate $${\tan }^{-1}\left(\frac{2a^x}{1-a^{2x}}\right),a>1,-\mathrm{\infty }<x<0$$ ?

Differentiate $${\sin }^{-1}\left\{\frac{\sqrt{1+x}+\sqrt{1-x}}{2}\right\},0<x<1$$ ?

Differentiate $${\tan }^{-1}\left(\frac{\sqrt{1+a^2{x}^2}-1}{ax}\right),x\ne 0$$ ?

Differentiate $${\tan }^{-1}\left(\frac{\sin x}{1+\cos x}\right),-\pi <x<\pi$$ ?

Differentiate $${\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$$ ?

Differentiate $${\cos }^{-1}\left(\frac{1-x^{2n}}{1+x^{2n}}\right),<x<\mathrm{\infty }$$ ?

Differentiate $${\sin }^{-1}\left(\frac{1-x^2}{1+x^2}\right)+{\sec }^{-1}\left(\frac{1+x^2}{1-x^2}\right),x\in R$$ ?

Differentiate $${\tan }^{-1}\left(\frac{a+x}{1-ax}\right)$$ ?

Differentiate  $${\tan }^{-1}\left(\frac{\sqrt{x}+\sqrt{a}}{1-\sqrt{xa}}\right)$$ ?

Differentiate $${\tan }^{-1}\left(\frac{a+b\tan x}{b-a\tan x}\right)$$ ?

Differentiate $${\tan }^{-1}\left(\frac{a+bx}{b-ax}\right)$$ ?

 Differentiate $${\tan }^{-1}\left(\frac{x-a}{x+a}\right)$$ ?

Differentiate $${\tan }^{-1}\left(\frac{x}{1+6x^2}\right)$$ ?

Differentiate $${\tan }^{-1}\left(\frac{5x}{1-6x^2}\right),-\frac{1}{\sqrt{6}}<x<\frac{1}{\sqrt{6}}$$ ?

Differentiate 

$${\tan }^{-1}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right),\frac{\pi }{4}<x<\frac{\pi }{4}$$ ?

Differentiate $${\tan }^{-1}\left\{\frac{x^{1/3}+a^{1/3}}{1-{\left(ax\right)}^{1/3}}\right\}$$ ?

Differentiate $${\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$$ with respect to x.

If  $$y={\sin }^{-1}\left(\frac{2x}{1+x^2}\right)+{\sec }^{-1}\left(\frac{1+x^2}{1-x^2}\right),0<x<1,$$ prove that  $$\frac{dy}{dx}=\frac{4}{1+x^2}$$ ?

If $$y={\sin }^{-1}\left(\frac{x}{1+x^2}\right)+{\cos }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right),0<x<\mathrm{\infty }$$ prove that  $$\frac{dy}{dx}=\frac{2}{1+x^2}$$ ?

Differentiate the following with respect to x: 

$${\cos }^{-1}(\sin x)$$

Differentiate the following with respect to x: 

$${\cot }^{-1}\left(\frac{1-x}{1+x}\right)$$

If  $$y={\cot }^{-1}\left\{\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right\}$$,  show that $$\frac{dy}{dx}$$ is independent of x. ? 

If $$y={\tan }^{-1}\left(\frac{2x}{1-x^2}\right)+{\sec }^{-1}\left(\frac{1+x^2}{1-x^2}\right),x>0$$ ,prove that $$\frac{dy}{dx}=\frac{4}{1+x^2}$$ ? 

If  $$y=se{c}^{-1}\left(\frac{x+1}{x-1}\right)+{\sin }^{-1}\left(\frac{x-1}{x+1}\right),x>0.\text{ Find}\frac{dy}{dx}$$ ?

If $$y=\sin [2{\tan }^{-1}\left\{\frac{\sqrt{1-x}}{1+x}\right\}],\text{ find }\frac{dy}{dx}$$ ?

If  $$y={\cos }^{-1}\left(2x\right)+2{\cos }^{-1}\sqrt{1-4x^2},0<x<\frac{1}{2},\text{ find }\frac{dy}{dx}.$$ ?

If the derivative of tan⁻¹ (a + bx) takes the value 1 at x = 0, prove that 1 + a² = b ?

If $$y={\cos }^{-1}\left(2x\right)+2{\cos }^{-1}\sqrt{1-4x^2},-\frac{1}{2}<x<0,\text{ find }\frac{dy}{dx}$$ ?

If $$y={\tan }^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right),\text{find }\frac{dy}{dx}$$ ?

If $$y={\cos }^{-1}\left\{\frac{2x-3\sqrt{1-x^2}}{\sqrt{13}}\right\},\text{ find }\frac{dy}{dx}$$ ?

Differentiate $${\sin }^{-1}\left\{\frac{2^{x+1}\cdot 3^x}{1+{\left(36\right)}^x}\right\}$$  with respect to x ?

If $$y={\sin }^{-1}\left(6x\sqrt{1-9x^2}\right),-\frac{1}{3\sqrt{2}}<x<\frac{1}{3\sqrt{2}}$$ $$\frac{dy}{dx}$$ ?

Find $$\frac{dy}{dx}$$ in the following case $$xy=c^2$$  ?

Find  $$\frac{dy}{dx}$$ in the following case: $$y^3-3x{y}^2=x^3+3x^{2}y$$ ?

Find  $$\frac{dy}{dx}$$ in the following case  $$x^{2/3}+y^{2/3}=a^{2/3}$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $$4x+3y=\log (4x-3y)$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $$x^5+y^5=5xy$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $${(x+y)}^2=2axy$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $${(x^2+y^2)}^2=xy$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $${\tan }^{-1}(x^2+y^2)=a$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $$e^{x-y}=\log \left(\frac{x}{y}\right)$$ ?

Find  $$\frac{dy}{dx}$$ in the following case $$\sin xy+\cos (x+y)=1$$ ?

If $$\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$$ , prove that $$\frac{dy}{dx}=\frac{\sqrt{1-y^2}}{1-x^2}$$ ?

If $$y\sqrt{1-x^2}+x\sqrt{1-y^2}=1$$ ,prove that $$\frac{dy}{dx}=-\sqrt{\frac{1-y^2}{1-x^2}}$$ ?

If $$xy=1$$ prove that $$\frac{dy}{dx}+y^2=0$$ ?

If $$x{y}^2=1,$$ prove that $$2\frac{dy}{dx}+y^3=0$$ ?

If $$x\sqrt{1+y}+y\sqrt{1+x}=0$$ , prove that $${(1+x)}^2\frac{dy}{dx}+1=0$$  ?

If $$\log \sqrt{x^2+y^2}={\tan }^{-1}\left(\frac{y}{x}\right)$$ Prove that $$\frac{dy}{dx}=\frac{x+y}{x-y}$$ ?

If $$\sec \left(\frac{x+y}{x-y}\right)=a$$ Prove that  $$\frac{dy}{dx}=\frac{y}{x}$$ ?

If $${\tan }^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)=a$$ Prove that  $$\frac{dy}{dx}=\frac{x}{y}\frac{(1-\tan a)}{(1+\tan a)}$$ ?

If $$xy\log (x+y)=1$$ ,Prove that $$\frac{dy}{dx}=-\frac{y(x^{2}y+x+y)}{x(x{y}^2+x+y)}$$ ?

If $$y=x\sin (a+y)$$ ,Prove that $$\frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin (a+y)-y\cos (a+y)}$$ ?

If $$x\sin (a+y)+\sin a\cos (a+y)=0$$ Prove that $$\frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin a}$$ ?

If $$y=x\sin y$$ , Prove that $$\frac{dy}{dx}=\frac{\sin y}{(1-x\cos y)}$$ ?

If $$y\sqrt{x^2+1}=\log (\sqrt{x^2+1}-x)$$ ,Show that $$(x^2+1)\frac{dy}{dx}+xy+1=0$$ ?

If $$\sin \left(xy\right)+\frac{y}{x}=x^2-y^2,\text{ find}\frac{dy}{dx}$$ ?

If  $$\tan (x+y)+\tan (x-y)=1,\text{ find}\frac{dy}{dx}$$ ?

If $$e^x+e^y=e^{x+y},\text{ prove that }\frac{dy}{dx}=-\frac{e^x(e^y-1)}{e^y(e^x-1)}or\frac{dy}{dx}+e^{y-x}=0$$ ?

If $$\cos y=x\cos (a+y),\text{ with }\cos a\ne \pm 1,\text{ prove that }\frac{dy}{dx}=\frac{{\cos }^2(a+y)}{\sin a}$$ ?

If $${\sin }^{2}y+\cos xy=k,$$ find  $$\frac{dy}{dx}$$ at $$x=1,$$ $$y=\frac{\pi }{4}.$$ 

If $$y=\{{\log }_{\cos x}\sin x\}{\{{\log }_{\sin x}\cos x\}}^{-1}+{\sin }^{-1}\left(\frac{2x}{1+x^2}\right),\text{ find }\frac{dy}{dx}\text{ at }x=\frac{\pi }{4}$$ ?

If $$\sqrt{y+x}+\sqrt{y-x}=c,\text{show that }\frac{dy}{dx}=\frac{y}{x}-\sqrt{\frac{y^2}{x^2}-1}$$ ?

Differentiate $$x^{1/x}$$  with respect to x.

Differentiate $$x^{\sin x}$$  ?

Differentiate $${(1+\cos x)}^x$$ ?

Differentiate $$x^{{\cos }^{-1}x}$$ ?

Differentiate $${(\log x)}^x$$ ?

Differentiate $${(\log x)}^{\cos x}$$ ?

Differentiate $${(\sin x)}^{\cos x}$$ ?

Differentiate $$e^{x\log x}$$ ?

Differentiate  $${(\sin x)}^{\log x}$$ ?

Differentiate $${10}^{\log \sin x}$$ ?

Differentiate $${(\log x)}^{\log x}$$ ?

Differentiate $${10}^{\left({10}^x\right)}$$ ?

Differentiate  $$\sin \left(x^x\right)$$ ?

Differentiate $${({\sin }^{-1}x)}^x$$ ?

Differentiate $$x^{{\sin }^{-1}x}$$  ?

Differentiate $${(\tan x)}^{1/x}$$ ?

Differentiate $$x^{{\tan }^{-1}x}$$  ?

Differentiate  $$\left(x^x\right)\sqrt{x}$$ ?

Differentiate $$x^{(\sin x-\cos x)}+\frac{x^2-1}{x^2+1}$$ ?

Differentiate  $$x^{x\cos x+}\frac{x^2+1}{x^2-1}$$  ?

Differentiate $${(x\cos x)}^x+{(x\sin x)}^{1/x}$$ ?

Differentiate$${(x+\frac{1}{x})}^x+x^{(1+\frac{1}{x})}$$ ?

Differentiate $$e^{\sin x}+{(\tan x)}^x$$ ?

Differentiate $${(\cos x)}^x+{(\sin x)}^{1/x}$$ ?

Differentiate  $$x^{x^2-3}+{(x-3)}^{x^2}$$ ?

Find  $$\frac{dy}{dx}$$ $$y=e^x+{10}^x+x^x$$ ?

find  $$\frac{dy}{dx}$$  $$y=\frac{{(x^2-1)}^3(2x-1)}{\sqrt{(x-3)(4x-1)}}$$ ?

Find  $$\frac{dy}{dx}$$  $$y=\frac{e^{ax}\cdot \sec x\cdot \log x}{\sqrt{1-2x}}$$ ?

Find  $$\frac{dy}{dx}$$ $$y=e^{3x}\sin 4x\cdot 2^x$$ ?

Find  $$\frac{dy}{dx}$$ $$y=\sin x\sin 2x\sin 3x\sin 4x$$ ?

Find  $$\frac{dy}{dx}$$ $$y=x^{\sin x}+{(\sin x)}^x$$ ?

Find  $$\frac{dy}{dx}$$  $$y={(\sin x)}^{\cos x}+{(\cos x)}^{\sin x}$$ ?

Find $$\frac{dy}{dx}$$ $$y={(\tan x)}^{\cot x}+{(\cot x)}^{\tan x}$$ ?

If ${\displaystyle y={\left(\sin x\right)}^x+{\sin }^{-1}\sqrt{x} \text{then find} \frac{dy}{dx}}$ 

Find $$\frac{dy}{dx}$$ $$y=x^{\cos x}+{(\sin x)}^{\tan x}$$ ?

Find $$\frac{dy}{dx}$$  $$y=x^x+{(\sin x)}^x$$ ?

Find $$\frac{dy}{dx}$$ $$y={(\tan x)}^{\log x}+{\cos }^2\left(\frac{\pi }{4}\right)$$ ?

Find $$\frac{dy}{dx}$$

$$y=x^x+x^{1/x}$$ ?

Find $$\frac{dy}{dx}$$ $$y=x^{\log x}+{(\log x)}^x$$ ?

If $$x^{13}{y}^7={(x+y)}^{20}$$ prove that $$\frac{dy}{dx}=\frac{y}{x}$$ ?

If $$x^{16}{y}^9={(x^2+y)}^{17}$$ ,prove that $$x\frac{dy}{dx}=2y$$ ?

If $$y=\sin \left(x^x\right)$$ prove that  $$\frac{dy}{dx}=\cos \left(x^x\right)\cdot x^x(1+\log x)$$ ?

If $$x^x+y^x=1$$, prove that $$\frac{dy}{dx}=-\left\{\frac{x^x(1+\log x)+y^x\cdot \log y}{x\cdot y^{(x-1)}}\right\}$$ ?

If $$x^y\cdot y^x=1$$ , prove that $$\frac{dy}{dx}=-\frac{y(y+x\log y)}{x(y\log x+x)}$$ ?

If $$x^y+y^x={(x+y)}^{x+y},\text{ find }\frac{dy}{dx}$$ ?

If $$x^m{y}^n=1$$ , prove that $$\frac{dy}{dx}=-\frac{my}{nx}$$ ?

If $$y^x=e^{y-x}$$ ,prove that $$\frac{dy}{dx}=\frac{{(1+\log y)}^2}{\log y}$$ ?

If $${(\sin x)}^y={(\cos y)}^x,$$, prove that $$\frac{dy}{dx}=\frac{\log \cos y-ycotx}{\log \sin x+x\tan y}$$ ?

If $${(\cos x)}^y={(\tan y)}^x$$ , prove that $$\frac{dy}{dx}=\frac{\log \tan y+y\tan x}{\log \cos x-x\sec ycosecy}$$ ?

If $$e^x+e^y=e^{x+y}$$ , prove that

$$\frac{dy}{dx}+e^{y-x}=0$$ ?

If $$e^y=y^x,$$ prove that$$\frac{dy}{dx}=\frac{{(\log y)}^2}{\log y-1}$$ ?

If $$e^{x+y}-x=0$$ ,prove that $$\frac{dy}{dx}=\frac{1-x}{x}$$ ?

If $$y=x\sin (a+y)$$ , prove that $$\frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin (a+y)-y\cos (a+y)}$$ ?

If  $$x\sin (a+y)+\sin a\cos (a+y)=0$$ , prove that $$\frac{dy}{dx}=\frac{{\sin }^2(a+y)}{\sin a}$$ ?

If  $${(\sin x)}^y=x+y$$ , prove that $$\frac{dy}{dx}=\frac{1-(x+y)y\cot x}{(x+y)\log \sin x-1}$$ ?

If $$xy\log (x+y)=1$$ , prove that  $$\frac{dy}{dx}=-\frac{y(x^{2}y+x+y)}{x(x{y}^2+x+y)}$$ ?

If $$y=x\sin y$$ , prove that  $$\frac{dy}{dx}=\frac{y}{x(1-x\cos y)}$$ ?