Question

Question #41397

Answer 1

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))`