Question

Differentiate the function; H(y)=(y^3+6y^2-11y+36)(y^2-y+17)

Answer 1

yuk. This just a product of two functions. So use the product rule. The product rule says if f and g are functions of x(or y in this case), then (fg)'=f'g+g'f where f and g and f' and g' are function in y.

Answer 2

Alternatively, you can go about this using logarithmic differentiation. Given a function \(y=f(x)g(x),\) you take the log of both sides and differentiate:
\[\ln(y)=\ln(f(x)g(x))\\
\ln(y)=\ln(f(x))+\ln(g(x))\\
\frac{y'}{y}=\frac{f'(x)}{f(x)}+\frac{g'(x)}{g(x)}\\
y'=f(x)g(x)\left[\frac{f'(x)}{f(x)}+\frac{g'(x)}{g(x)}\right]\]
This method may be handier if you're given the product of more than two functions that have simple derivatives. For example, \(\large \displaystyle y=(x+9)(x-8)(x+7)(x-6)(\cdots)\)