Differentiate the following function.
y ' = ?
ATTEMPT
y=x^1/2(5x-2)
= 5x^(3/2)-2x(1/2)
= (15/2)x^(1/2)-x^(-1/2)
Then from this step I am lost. So please help me out! THANKS!
You have done it correctly. Although it is bad form to write it the way you've written it. You should have:
$\displaystyle y = \sqrt{x} (5x - 2) $
$\displaystyle y = x^{\frac{1}{2}} (5x - 2) $
$\displaystyle y = 5x^{\frac{3}{2}} - 2x^{\frac{1}{2}} $
$\displaystyle \therefore y' = \frac{15}{2} x^{\frac{1}{2}} - x^{\frac{-1}{2}} $
Now, you can make it look a bit prettier from here. Remember that $\displaystyle x^{-a} = \frac{1}{x^{a}} $. So we can write:
$\displaystyle y' = \frac{15}{2} x^{\frac{1}{2}} - \frac{1}{x^{\frac{1}{2}}} $
Now write it with square root signs!
$\displaystyle y' = \frac{15}{2} \sqrt{x} - \frac{1}{\sqrt{x}} $.
An easier way to do this might been with the product rule, which states that if $\displaystyle y = f(x) g(x) $, then $\displaystyle y' = f'(x) g(x) + f(x) g'(x) $