find dy/dx by implicit differentation and evaluate the derivitave at the given point: (x+y)^3:x^3+y^3 point=(-1,1)
To find dy/dx by implicit differentiation, follow these steps:
Step 1: Write the given equation in the form F(x, y) = 0. In this case, the equation is (x + y)^3 = x^3 + y^3.
Step 2: Differentiate both sides of the equation with respect to x using the chain rule for the term (x + y)^3.
d/dx[(x + y)^3] = d/dx[x^3 + y^3]
Step 3: Apply the chain rule to the left side of the equation:
3(x + y)^2 * (1 + dy/dx) = 3x^2 + 3y^2 * dy/dx
Step 4: Rearrange the equation to solve for dy/dx:
3(x + y)^2 * dy/dx = 3x^2 + 3y^2 - 3(x + y)^2
dy/dx = (3x^2 + 3y^2 - 3(x + y)^2) / (3(x + y)^2)
Step 5: Evaluate the derivative at the given point (-1, 1) by substituting the values of x and y into the derivative expression:
dy/dx = (3(-1)^2 + 3(1)^2 - 3(-1 + 1)^2) / (3(-1 + 1)^2)
= (3 + 3 - 3(0)^2) / (3(0)^2)
= 6/0
The derivative at the given point is undefined (dy/dx = 6/0).