Derivative of $ \sqrt y + \sqrt x = 4 $ at $ ( 0.25 , 0.25 ) $

Question

Find the derivative of $ \sqrt y + \sqrt x = 4 $ at $ ( 0.25 , 0.25 ) $.

Finding derivative, I get $ \frac { \mathrm d y } { \mathrm d x } = - \sqrt { \frac y x } $.

At $ ( 0.25 , 0.25 ) $, the value is $ - 1 $.

But using another method,

$$ \sqrt y = 4 - \sqrt x $$ $$ y = ( 4 - \sqrt x ) ^ 2 $$ $$ \frac { \mathrm d y } { \mathrm d x } = 1 - \frac 4 { \sqrt x } $$ At $ ( 0.25 , 0.25 ) $, the value is $ - 7 $.

Which is the correct answer? Is there any error in this method?

Answer 1

Yes, you are must in error, because $(0.25,0.25)$ is not lie on curve $\sqrt x+\sqrt y=4$, but lie on $\sqrt x+\sqrt y=1$.

In this case, we can calculate it by two way that you show, and it is exactly same number.

Case 1: $\frac{dy}{dx}=-\sqrt{\frac yx}=-1$.

Case 2: $y=(1-\sqrt x)^2=1-2\sqrt x+x$, so $\frac{dy}{dx}=1-\frac1{\sqrt x}=-1$.

Answer 2

Let us consider two cases.

Case 1) The curve is $$\sqrt y + \sqrt x = 4$$ and the point is $(x,y)=(4,4)$ $$\sqrt y + \sqrt x = 4 \implies \frac {\frac {dy}{dx} }{2\sqrt y}+\frac {1}{2\sqrt x}=0$$

We get $$\frac {dy}{dx}(4,4)=-1$$

Case 2) The curve is $$\sqrt y + \sqrt x = 1$$ and the point is $(x,y)=(0.25,0.25)$ $$\sqrt y + \sqrt x = 1 \implies \frac {\frac {dy}{dx} }{2\sqrt y}+\frac {1}{2\sqrt x}=0$$ We get $$\frac {dy}{dx}(.25,.25)=-1$$ We get the same result in both cases.

Answer 3

Note that, if the equation is $\sqrt x+\sqrt y=4$, then we have

$$1-{4\over\sqrt x}=1-{\sqrt x+\sqrt y\over\sqrt x}=-{\sqrt y\over\sqrt x}$$

so your two computations of $dy/dx$ agree. As others have pointed out, the error is that the point $(x,y)=(0.25,0.25)$ do not lie on the curve $\sqrt x+\sqrt y=4$, so it makes no sense to talk about the derivative there -- i.e., since the point is not on the curve, the curve has no tangent line at that point, and it makes no sense to ask for the slope of a tangent line that doesn't exist.

There are two sensible ways to correct things: either change the equation to $\sqrt x+\sqrt y=1$, or change the point to $(x,y)=(4,4)$. Either way you fix things, the derivative will be $-1$, which can be understood from the formal symmetry of interchanging $x$ and $y$.