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can someone explain this step (from implicit differentiation question)

Attachment 25387

I'm not sure how the numerator became 12x^3/2 -1

Re: can someone explain this step (from implicit differentiation question)

Hello, kingsolomonsgrave!

Quote:

$\displaystyle \frac{dy}{dx} \;=\;\frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} \quad\Rightarrow\quad \frac{dy}{dx} \;=\; \frac{12x^{\frac{3}{2}}-1}{2\sqrt{x}\sin(y)}\quad\text{How?}$

We have: .$\displaystyle \frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} $

Multiply by$\displaystyle \frac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}\!:\;\; \frac{2x^{\frac{1}{2}}}{2x^{\frac{1}{2}}}\cdot \frac{6x-\frac{1}{2}x^{-\frac{1}{2}}}{\sin(y)} \;=\;\frac{12x^{\frac{3}{2}} - 1}{2x^{\frac{1}{2}}\sin(y)} $

Re: can someone explain this step (from implicit differentiation question)

thanks! we know to choose 2x^1/2/2x^1/2 because that will make the 1/2x^-1/2 equal to 1?