Find the derivative of implicit function defined by $x^2 + \sqrt{xy} = 7$

Question

Assuming that the equation determines a differentiable function $f$ such that $y=f(x)$, then find $y'$.

$$x^2 + \sqrt{xy} = 7$$

I think I should use implicit differentiation, but how?

Answer 1

Differentiating both sides with respect to $x$ we get

$$2x+\frac{1}{2\sqrt{xy}}(y+xy') = 0$$

(Remember to use the chain rule and the product rule when differentiating $\sqrt{xy}$)

Now solve for $y'$ (first multiply both sides by $2\sqrt{xy}$ to simplify):

$$y'=\frac{-4x\sqrt{xy}-y}{x}$$

or

$$y'=-4\sqrt{xy}-\frac{y}{x}$$