Question

Find the derivative of

\(f (x) = x^{\frac{1}{3}} \left( 10 x^3 + 41 x^2 + 10 \right)\)

The only value of \(x\) for which the derivative is undefined is \(x =\) ?

Answer 1

f'x = 1/3x^4/3(10x^3+41x^2+10).(30x^2+82)

Answer 2

You can do this in two ways. The first way is to multiply \(x^{\frac{1}{3}}\) inside the parenthesis and then use the power rule. The second way is by using the product rule. For me, the first way is better.
\(f(x)=10x^{3+\frac{1}{3}}+41x^{2+\frac{1}{3}}+10x^{\frac{1}{3}}=10x^{\frac{10}{3}}+41x^{\frac{7}{3}}+10x^{\frac{1}{3}}\).
\(f' (x)=\frac{10}{3}x^{\frac{7}{3}}+\frac{287}{3}x^{\frac{4}{3}}+\frac{10}{3}x^{-\frac{2}{3}}\).