Question #41397

1 Answer
Oct 2, 2017

After using logarithmic differentiation, the answer ends up being dy/dx=(ln(x)^{cos(10x)}*(cos(10x)-10x sin(10x)ln(x)ln(ln(x))))/(x ln(x))

Explanation:

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

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

1/y * dy/dx=-10sin(10x)ln(ln(x))+cos(10x) * 1/(ln(x)) * 1/x

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

dy/dx=(ln(x)^{cos(10x)}*(cos(10x)-10x sin(10x)ln(x)ln(ln(x))))/(x ln(x))