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Section 3-9 : Chain Rule

11. Differentiate $$F\left( y \right) = \ln \left( {1 - 5{y^2} + {y^3}} \right)$$ .

Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule.
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For this problem the outside function is (hopefully) clearly the logarithm and the inside function is the stuff inside of the logarithm. The derivative is then,

$F\left( y \right) = \frac{1}{{1 - 5{y^2} + {y^3}}}\left( { - 10y + 3{y^2}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{ - 10y + 3{y^2}}}{{1 - 5{y^2} + {y^3}}}}}$

With logarithm problems remember that after differentiating the logarithm (i.e. the outside function) you need to substitute the inside function into the derivative. So, instead of getting just,

$\frac{1}{y}$

we get the following (i.e. we plugged the inside function into the derivative),

$\frac{1}{{1 - 5{y^2} + {y^3}}}$

Then, we can’t forget of course to multiply by the derivative of the inside function.