Derivative using Product Rule I have a derivative:
$$ \frac{d}{dx}[\sqrt{10x}]$$
I start by using $ ^n\sqrt{a^{x}}=a^\frac{x}{n}$ to give me:
$$\frac{d}{dx}[(10x)^\frac{1}{2}]$$
I can factor out 10 to give me:
$$\frac{d}{dx}[(10(x))^\frac{1}{2}]$$
This is the part that loses me.  I know I need to use the Product Rule to $10(x)$, but I must not be setting it up correctly.  When I use the Product Rule I get:
$$(f*g)'=f'*g+f*g'$$
$$(f*g)'=(10)'*(x)+(10)*(x)'$$
$$(f*g)'=0*(x)+(10)*1$$
$$(f*g)'=10$$
What I've been told the result should be after using the Product Rule is:
$$\frac{d}{dx}[10^\frac{1}{2}x^\frac{1}{2}]$$
Where would I be going wrong when it comes to using the Product Rule?
 A: When you are multiplying two functions and one of them is just a constant, then when you take the derivative, you can just pull the constant out.  No need for the product rule in that case.
For this problem, you don't need the chain rule either, just the power rule!
$$ \frac{d}{dx}[10x]^{1/2} = 10^{1/2} \frac{d}{dx} x^{1/2} = 10^{1/2}\cdot \frac{1}{2} \cdot x^{-1/2} = \frac{\sqrt{10}}{2 \sqrt{x}}$$
If you want to use the chain rule instead, then $10x$ is your inside function and $\sqrt{x}$ is your outside function.
A couple of points: you wrote $\frac{d}{dx} = [10x]^{1/2}$ when what you mean is $\frac{d}{dx} [10x]^{1/2}$.  That is not an equation, it is an expression, the derivative operator, acting on a function.
Also, your final result does not have calculus done, it was just the bit of algebra that says $[10x]^{1/2} = 10^{1/2} x^{1/2}$.  Maybe you understood that, but your first mistake makes it a little hard to tell.
A: Since constants have a zero derivative, don't bother applying the product rule to them, just factor them out.
i.e. $(c\times f(x))' = 0\times f(x)+c\times f'(x)=c\times f'(x)$ is the same as $c\times (f(x))'$ where $c$ was factored out.

*

*by factoring out the constant

$(\sqrt{10x})'=(\sqrt{10}\times\sqrt{x})'=\sqrt{10}\times (\sqrt{x})'=\sqrt{10}\times (x^\frac 12)'=\sqrt{10}\times \frac 12x^{-\frac 12}=\dfrac{\sqrt{10}}{2\sqrt{x}}$

*

*by chain rule

$\require{cancel}((10x)^\frac 12)'=\frac 12(10x)^{-\frac 12}\times (10x)'=\frac 12(\cancel{10}x)^{-\frac 12}\times \underbrace{10}_{10^\frac 12\cdot\cancel{10^\frac 12}}=\frac 1210^\frac12x^{-\frac12}=\dfrac{\sqrt{10}}{2\sqrt{x}}$
And this is the same result.
