Product Rule Shortcut for Complicated Derivatives

[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?

 

[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.

 

[statdad]

You could take logs of each side:
$$\begin{align*}<br /> \ln y & = \ln{\left(x^2+1\right)^7 \left(x^9+2\right)^5 \left(x^3+1\right)^3 \left(x^8 + 7\right)^3} \\<br /> & = 7\ln(x^2+1) + 5\ln(x^9+1) + 3\ln(x^3+1) + 3\ln(x^8+7)<br /> \end{align*}$$

Differentiating gives
$$\frac{y'}{y}$$

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.