How do you find the derivative of $x \cdot x^{\frac{1}{2}}$ $x*x^(1/2)$ ?

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How do you find the derivative of $x \cdot x^{\frac{1}{2}}$ $x*x^(1/2)$ ?

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Answer 1 · 2015-08-08T22:27:14

$\frac{d}{d x} (x \cdot x^{\frac{1}{2}}) = \frac{d}{d x} x^{\frac{3}{2}} = \frac{3}{2} x^{\frac{1}{2}}$ $d/(dx) (x*x^(1/2)) = d/(dx) x^(3/2) = 3/2x^(1/2)$

or

$\frac{d}{d x} (x \cdot x^{\frac{1}{2}}) = (\frac{d}{d x} x) \cdot x^{\frac{1}{2}} + x \cdot (\frac{d}{d x} x^{\frac{1}{2}})$ $d/(dx) (x*x^(1/2)) = (d/(dx) x) * x^(1/2) + x * (d/(dx)x^(1/2))$

$= 1 \cdot x^{\frac{1}{2}} + x \cdot (\frac{1}{2}) x^{- \frac{1}{2}} = x^{\frac{1}{2}} + \frac{1}{2} x^{\frac{1}{2}} = \frac{3}{2} x^{\frac{1}{2}}$ $=1*x^(1/2)+x*(1/2)x^(-1/2) = x^(1/2)+1/2 x^(1/2) = 3/2 x^(1/2)$

Explanation:

We can either multiply $x \cdot x^{\frac{1}{2}} = x^{\frac{3}{2}}$ $x*x^(1/2) = x^(3/2)$ first then use the power rule, or we can use the product rule, using the power rule on each part.

Power Rule
$\frac{d}{d x} x^{k} = k \cdot x^{k - 1}$ $d/(dx) x^k = k*x^(k-1)$

Product Rule
$d/(dx) (f(x)*g(x)) = f'(x)g(x)+f(x)g'(x)$

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How do you find the derivative of $x \cdot x^{\frac{1}{2}}$ $x*x^(1/2)$ ?

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