Question

Differentiate ${\left(\cos x\right)}^{\sin x}$ with respect to ${\left(\sin x\right)}^{\cos x}$.

Answer 1

$\mathrm{Let},u={\left(\cos x\right)}^{\sin x}$

Taking log on both sides,

$$\begin{gathered} \log u=\log {\left(\cos x\right)}^{\sin x} \\ \Rightarrow \log u=\sin x\log \left(\cos x\right) \end{gathered}$$

Differentiating it with respect to x using chain rule,

$$\begin{gathered} \frac{1}{u}\frac{du}{dx}=\sin x\frac{d}{dx}\left(\log \cos x\right)+\log \cos x\frac{d}{dx}\left(\sin x\right)\left[\mathrm{using}\mathrm{product}\mathrm{rule}\right] \\ \Rightarrow \frac{1}{u}\frac{du}{dx}=\sin x\left(\frac{1}{\cos x}\right)\frac{d}{dx}\left(\cos x\right)+\log \cos x\left(\cos x\right) \\ \Rightarrow \frac{du}{dx}=u\left[\left(\tan x\right)\times \left(-\sin x\right)+\log \log x\left(\cos x\right)\right] \\ \Rightarrow \frac{du}{dx}={\left(\cos x\right)}^{\sin x}\left[\cos x\log \cos x-\sin x\tan x\right]...\left(\mathrm{i}\right) \\ \mathrm{Let},v={\left(\sin x\right)}^{\cos x} \end{gathered}$$

Taking log on both sides,

$$\begin{gathered} \log v=\log {\left(\sin x\right)}^{\cos x} \\ \Rightarrow \log v=\cos x\log \left(\sin x\right) \end{gathered}$$

Differentiating it with respect to x using chain rule,

$$\begin{gathered} \frac{1}{v}\frac{dv}{dx}=\cos x\frac{d}{dx}\left(\log \sin x\right)+\mathrm{logsin}x\frac{d}{dx}\left(\cos x\right)\left[\mathrm{using}\mathrm{product}\mathrm{rule}\right] \\ \Rightarrow \frac{1}{v}\frac{dv}{dx}=\cos x\left(\frac{1}{\sin x}\right)\frac{d}{dx}\left(\sin x\right)+\log \sin x\left(-\sin x\right) \\ \Rightarrow \frac{dv}{dx}=v\left[cotx\left(\cos x\right)-\sin x\log \sin x\right] \\ \Rightarrow \frac{dv}{dx}={\left(\sin x\right)}^{\cos x}\left[cotx\left(\cos x\right)-\sin x\log \sin x\right] \\ \mathrm{dividing}\mathrm{equation}\left(\mathrm{i}\right)\mathrm{by}\left(\mathrm{ii}\right), \\ \therefore \frac{du}{dv}=\frac{{\left(\cos x\right)}^{\sin x}\left[\cos x\log \cos x-\sin x\tan x\right]}{{\left(\sin x\right)}^{\cos x}\left[cotx\left(\cos x\right)-\sin x\log \sin x\right]} \end{gathered}$$