Question

${10}^{{10}^{{\log }_{10}\csc \left(10x\right)}}$ differentiate w.r.t. $x$

Answer 1

Solution:-

${10}^{{10}^{{\log }_{10}\csc 10x}}$

Differentiating w.r.t. $x$, we have

$\frac{dy}{dx}\left({10}^{{10}^{{\log }_{10}\csc 10x}}\right)$

Differentiating according to chain rule, we have

$=\left({10}^{{10}^{{\log }_{10}\csc 10x}}\log 10\right)\times \left({10}^{{\log }_{10}\csc 10x}\log 10\right)\times (\frac{1}{\csc 10x}\times (-csc10x\cot 10x.)\times 10)$

$\Rightarrow \frac{dy}{dx}\left({10}^{{10}^{{\log }_{10}\csc 10x}}\right)=-\left({10}^{{10}^{{\log }_{10}\csc 10x}}\right)\times \left({10}^{{\log }_{10}\csc 10x}\right)\times (\cot 10x\times 10)$

Hence $\frac{dy}{dx}=-10\cot \left(10x\right)\left({10}^{{10}^{{\log }_{10}\csc 10x}}\right)\times \left({10}^{{\log }_{10}\csc 10x}\right)$