Answer
$y=\frac{3x}{4}-6$
Work Step by Step
Original Equation:
$(x-1)^{2}+(y-1)^2=25$
Differentiate Implicitly Using Chain Rule:
$2(x-1)\frac{dx}{dx}+2(y-1)\frac{dy}{dx}=0$
Simplify:
$(2x-2)+(2y-2)\frac{dy}{dx}=0$
$(2y-2)\frac{dy}{dx}=-2x+2$
$\frac{dy}{dx}=\frac{-2x+2}{2y-2}$
$\frac{dy}{dx}=\frac{-x+1}{y-1}$
Plug in $(x,y)$
$\frac{dy}{dx}=\frac{3}{4}$
Point Slope form:
$y-y_{1}=\frac{dy}{dx}(x-x_{1})$
Plug in Values:
$y+3=\frac{3}{4}(x-4)$
Simplify:
$y=\frac{3x}{4}-6$
