Help checking my problem. I have some idea what to do. Find the slope of the line tangent of the curve defined implicitly by x^2y + (xy)^(1/2) = 36
nicer looking equation inside.

Question

Help checking my problem. I have some idea what to do. Find the slope of the line tangent of the curve defined implicitly by x^2y + (xy)^(1/2) = 36
nicer looking equation inside.

anonymous · Answer

$$x^2y+\sqrt{xy}=36$$

solomonzelman · Answer

slope of the tangent line at x what?

anonymous · Answer

oh whoops at (2,8)

solomonzelman · Answer

so at x=2.

solomonzelman · Answer

first, you have to take the derivative (you know that)

can you derive it?
(Keep in mind the chain rule, multiplying times  dy/dx  every time you get a derivative of y)

solomonzelman · Answer

can you differentiate    x^2 y    first?

anonymous · Answer

that becomes:
$$2xy + x^2\frac{ dy }{ dx }$$

solomonzelman · Answer

yes, correct.

solomonzelman · Answer

now the square root of  $\large\color{red}{  \sqrt{xy}  }$.
keep in mind the chain rule, not only for the product of x and y, but for y (the dy/dx) .

solomonzelman · Answer

Saying that.
We know that  $\large\color{red}{ \frac{d}{dx} \sqrt{x} =\frac{1}{2\sqrt{x}} }$

and We know that  $\large\color{blue}{ \frac{d}{dx} xy =y+x\frac{dy}{dx} }$

solomonzelman · Answer

So, the square root of (xy) does as follows,
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