Question

derivative y = sqrt(3x+2)/(csc^2(x^3+1) solve with logarithmic differentiation

Answer 1

Show me your best attempt. Remember, square root is the same thing as to the 1/2 power.

Answer 2

[(1/2\ln(3x+2)) / 2lncsc(x^3+1)] im not sure what to do after this

Answer 3

There's a mistake with the log property you used..
\[\begin{align*}y&=\frac{\sqrt{3x+2}}{\csc^2(x^3+1)}\\
\ln y&=\ln\frac{\sqrt{3x+2}}{\csc^2(x^3+1)}\\
\ln y&=\ln(3x+2)^{1/2}-\ln\csc^2(x^3+1)\\
\ln y&=\frac{1}{2}\ln(3x+2)-2\ln\csc(x^3+1)\end{align*}\]
Next step would be implicit differentiation.

Answer 4

why did the division between the 3x+2 and the scs disappear?

Answer 5

That's the property: log of a quotient is becomes the difference of logs. Mathematically, that's
\[\ln\frac{a}{b}=\ln a-\ln b\]