# Question #41397

Oct 2, 2017

After using logarithmic differentiation, the answer ends up being $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\ln {\left(x\right)}^{\cos \left(10 x\right)} \cdot \left(\cos \left(10 x\right) - 10 x \sin \left(10 x\right) \ln \left(x\right) \ln \left(\ln \left(x\right)\right)\right)}{x \ln \left(x\right)}$

#### Explanation:

First take the log of both sides and use a property of logarithms to write $\ln \left(y\right) = \cos \left(10 x\right) \ln \left(\ln \left(x\right)\right)$.

Next, differentiate both sides with respect to $x$, keeping in mind that $y$ is a function of $x$ (the is the "logarithmic differentiation" part), and using the Chain Rule and Product Rule to get

$\frac{1}{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}} = - 10 \sin \left(10 x\right) \ln \left(\ln \left(x\right)\right) + \cos \left(10 x\right) \cdot \frac{1}{\ln \left(x\right)} \cdot \frac{1}{x}$

Finally, multiply both sides by $y = \ln {\left(x\right)}^{\cos \left(10 x\right)}$ and get a common denominator of $x \ln \left(x\right)$ to get the final answer:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\ln {\left(x\right)}^{\cos \left(10 x\right)} \cdot \left(\cos \left(10 x\right) - 10 x \sin \left(10 x\right) \ln \left(x\right) \ln \left(\ln \left(x\right)\right)\right)}{x \ln \left(x\right)}$