Use implicit differentiation to find the slope of the tangent line to the curve sqrt of x + sqrt of y = 8
x^.5 + y^.5 = 8
.5 x^-.5 + .5 y^-.5 dy/dx = 0
dy/dx = - sqrt x/ sqrt y
To find the slope of the tangent line to the curve sqrt(x) + sqrt(y) = 8 using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x:
d/dx(sqrt(x) + sqrt(y)) = d/dx(8)
Step 2: Apply the chain rule to differentiate the square root terms:
1/(2√x) * d(sqrt(x))/dx + 1/(2√y) * d(sqrt(y))/dx = 0
Step 3: Simplify and solve for d(sqrt(y))/dx, which represents dy/dx:
1/(2√x) * (1/2)(1/√x) + 1/(2√y) * dy/dx = 0
1/(4x√x) + 1/(2√y) * dy/dx = 0
Step 4: Multiply through by 2√y to isolate dy/dx:
2/√y * (1/(4x√x) + 1/(2√y) * dy/dx) = 0
1/(2√x) + 1/(√y) * dy/dx = 0
1/(√x) + 2/(√y) * dy/dx = 0
Step 5: Solve for dy/dx:
dy/dx = - (√y)/(2√x)
This is the derivative of y with respect to x, or the slope of the tangent line to the curve at any point (x, y) satisfying the equation of the curve.