find the slope of the tangent to the curve:
y^2=x^3/xy
To find the slope of the tangent to the curve, we need to take the derivative of the given curve equation. Here's how you can do it step by step:
Step 1: Rewrite the given equation in a simplified form:
y^2 = x^3 / (xy)
Step 2: Simplify the right-hand side of the equation:
y^2 = x^2 / y
Step 3: Take the derivative of both sides of the equation with respect to x (using the quotient rule):
2yy' = (2x * y - x^2 * y') / y^2
Step 4: Multiply both sides of the equation by y^2 to eliminate the fractions:
2yy'^3 = 2xy - x^2y'
Step 5: Rearrange the equation to solve for y':
2yy'^3 + x^2y' - 2xy = 0
Step 6: Factor out y':
y' (2y^2 + x^2) = 2xy
Step 7: Divide both sides of the equation by (2y^2 + x^2):
y' = 2xy / (2y^2 + x^2)
Therefore, the slope of the tangent to the curve is given by the equation y' = 2xy / (2y^2 + x^2).