how do you find the dy/dx of the square root of x minus the square root of y equal to 10? You have to use implicit differentiation.
sqrt(x-sqrty)=10
square both sides
x-sqrt y=100
dx-1/2 dy/sqrty =0
dx/dy= 1/2sqrty
To find the derivative \( \frac{{dy}}{{dx}} \) using implicit differentiation, follow these steps:
1. Start with the equation \( \sqrt{x} - \sqrt{y} = 10 \).
2. Differentiate each term with respect to \( x \). For \( \sqrt{x} \), apply the power rule \( \frac{{d}}{{dx}}(\sqrt{x}) = \frac{1}{2}x^{-\frac{1}{2}} \) and for \( \sqrt{y} \), use the chain rule \( \frac{{d}}{{dx}}(\sqrt{y}) = \frac{1}{2y^{-\frac{1}{2}}} \frac{{dy}}{{dx}} \):
\( \frac{1}{2x^{-\frac{1}{2}}} - \frac{1}{2y^{-\frac{1}{2}}} \frac{{dy}}{{dx}} = 0 \).
3. Simplify the expression:
\( \frac{1}{2\sqrt{x}} - \frac{1}{2\sqrt{y}} \frac{{dy}}{{dx}} = 0 \).
4. Isolate \( \frac{{dy}}{{dx}} \) by moving the second term to the right side of the equation:
\( \frac{1}{2\sqrt{y}} \frac{{dy}}{{dx}} = \frac{1}{2\sqrt{x}} \).
5. Multiply both sides of the equation by \( 2\sqrt{y} \) to eliminate the fraction:
\( \frac{{dy}}{{dx}} = \frac{{\sqrt{y}}}{{\sqrt{x}}} \).
So, the derivative \( \frac{{dy}}{{dx}} \) of \( \sqrt{x} - \sqrt{y} = 10 \) is \( \frac{{\sqrt{y}}}{{\sqrt{x}}} \).