I need to find the derivative of the function:
$\displaystyle f(x)= x + 1 / \sqrt{x}$
/ = divided by
Can someone show me step by step how I'd go about solving down? And if possible guide me on the steps they've used? Thanks!
$\displaystyle
\frac{d}
{{dx}}\left( {x + \frac{1}
{{\sqrt x }}} \right) = \frac{d}
{{dx}}\left( {x + x^{ - 1/2} } \right)\underbrace = _{{\text{sum rule}}}\frac{d}
{{dx}}\left( x \right) + \frac{d}
{{dx}}\left( {x^{ - 1/2} } \right)
$
But $\displaystyle
\frac{d}
{{dx}}x^q = q \cdot x^{q - 1} ,{\text{ }}\forall q \in \mathbb{Q}
$
Therefore: $\displaystyle
\frac{d}
{{dx}}\left( x \right) + \frac{d}
{{dx}}\left( {x^{ - 1/2} } \right) = 1 \cdot x^0 + \frac{{ - 1}}
{2} \cdot x^{\frac{{ - 1}}
{2} - 1} = 1 + \frac{{ - 1}}
{2} \cdot x^{ - 3/2} = 1 + \frac{1}
{{\sqrt {x^3 } }}
$