# Math Help - differentiation

1. ## differentiation

Use logarithmic differentiation to find the derivative of the following function:
y
= (x^3 + 1/(sin^2(x) · (x^5x^2)^7))^(1/4)

2. Originally Posted by Sally_Math
Use logarithmic differentiation to find the derivative of the following function:
y
= (x^3 + 1/(sin^2(x) · (x^5x^2)^7))^(1/4)

$y = \left[\frac{x^3+1}{\sin^2{x} \cdot (x^5-x^2)^7}\right]^{\frac{1}{4}}$

this more of an algebra drill than it is an exercise in derivatives. use the laws of logs to break down the expression ...

$\ln{y} = \frac{1}{4}\left[\ln(x^3+1) - 2\ln(\sin{x}) - 7\ln(x^5-x^2)\right]$

now take the derivative ...

$\frac{y'}{y} = \frac{1}{4}\left[\frac{3x^2}{x^3+1} - 2\cot{x} - \frac{7(5x^4-2x)}{x^5-x^2}\right]
$

finish up by multiplying both sides by y