Cem Hayes

2021-09-27

Determine the Derivatives
$f\left(x\right)=\mathrm{ln}\left[{e}^{\mathrm{sin}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)}\right]$

Brighton

Solution:
Given: The function is $f\left(x\right)=\mathrm{ln}\left[{e}^{\mathrm{sin}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)}\right]$.
Rewrite the function:
$f\left(x\right)=\mathrm{sin}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)$
Differentiate f with respect to x:
$\frac{d}{dx}f\left(x\right)=\mathrm{cos}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\frac{d}{dx}\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)$
$=\mathrm{cos}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\left(-{\mathrm{csc}h}^{2}\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\frac{d}{dx}\left(\mathrm{sin}h\left({x}^{2}\right)\right)$
$=-{\mathrm{csc}h}^{2}\left(\mathrm{sin}h\left({x}^{2}\right)\right)\mathrm{cos}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\mathrm{cos}h\left({x}^{2}\right)\frac{d}{dx}\left({x}^{2}\right)$
$=-{\mathrm{csc}h}^{2}\left(\mathrm{sin}h\left({x}^{2}\right)\right)\mathrm{cos}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\mathrm{cos}h\left({x}^{2}\right)\left(2x\right)$
Conclusion:
Therefore, the derivative is ${f}^{\prime }\left(x\right)=-2x{\mathrm{csc}h}^{2}\left(\mathrm{sin}h\left({x}^{2}\right)\right)\mathrm{cos}h\left(\mathrm{cot}h\left(\mathrm{sin}h\left({x}^{2}\right)\right)\right)\mathrm{cos}h\left({x}^{2}\right)$.

Jeffrey Jordon