# Product Chain Power Rule： Either it's the book or I am wrong.

The problem:　$x^3\sqrt{2x+4}$

$f(x):= x^3$, $g(x):= \sqrt{2x+4}$

$(f\times g)' = f^{\prime}g+fg^{\prime}$ thus it should be

$3x^2\sqrt{2x+4} + (x^3)[\frac{1}{2}(2x+4)^{\frac{-1}{2}}(2)]$

which is $3x^2\sqrt{2x+4}+\frac{x^3}{\sqrt{2x+4}}$

The book gives: $3x^2\sqrt{2x+4}+\frac{x^3}{2\sqrt{2x+4}}$

I'm correct? I always get worry when my answers don't match the book.

$\frac{d}{dx}[f\times g(h(x))] = f^{\prime} \times g(h(x))+ f\times g^{\prime}(h(x))h^{\prime}$ right?

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Last formula should be $g'(h(x))h'$, not $g(h(x))'h'$. – Arturo Magidin Feb 21 '12 at 2:55
Your answer to the problem is the correct one, I think. Books have typos. For simple derivatives such as this you can check with a computer, if you trust computers. – Dylan Moreland Feb 21 '12 at 2:55
@DylanMoreland I really know that I am correct, its just that I panic when I don't have what the book says. – yiyi Feb 21 '12 at 2:57
@Dylan You could write an answer so that OP will choose the answer and keep this away from those Unanswered list. – user21436 Feb 21 '12 at 3:33

As my colleagues have astutely pointed out, the product rule states $(fg)^{\prime} = f^{\prime} g + f g^{\prime}$. Define $f(x)= x^3$ and $g(x)= \sqrt{2x+4}$. As $f^{\prime}(x) = 3 x^{2}$ and $g^{\prime}(x) = \frac{1}{\sqrt{2x+4}}$, the product rule gives \begin{align} (f(x)g(x))^{\prime} = f^{\prime}(x) g(x) + f(x) g^{\prime}(x) = 3x^{2} \sqrt{2x + 4} + \frac{x^3}{\sqrt{2x+4}}. \end{align} The answer in your book has an incorrect factor of $2$ in the denominator of the second term. This factor should cancel with the factor of $2$ coming from $2x$ (in $g(x)$) by the chain rule.