# Derivatives - Product Rule

1. Feb 3, 2014

### tsaitea

Find y'
y=(x2+1)7(x9+2)5(x3+1)3(x8+7)3

Is there a shortcut to doing this problem? Or do I have to actually use the product rule more than 3 times?

2. Feb 3, 2014

### rock.freak667

The easiest way would be to just use the product rule. Using the chain rule might make it a bit easier in terms of making sure that you don't make a mistake.

3. Feb 3, 2014

\begin{align*} \ln y & = \ln{\left(x^2+1\right)^7 \left(x^9+2\right)^5 \left(x^3+1\right)^3 \left(x^8 + 7\right)^3} \\ & = 7\ln(x^2+1) + 5\ln(x^9+1) + 3\ln(x^3+1) + 3\ln(x^8+7) \end{align*}
$$\frac{y'}{y}$$
on the left and a sum of terms on the right: multiply through by $y$ and cancelling terms may make the work slightly more palatable.