Question

Find the slope of the tangent line to the curve √(8x) - 7y√(xy) = 32.

a) 1/8
b) 2/7
c) -1/7
d) -8/7

Answer 1

To find the slope of the tangent line to the curve, differentiate the equation implicitly and solve for the coordinates of the point where the tangent line intersects the curve. Finally, plug the coordinates into the derivative of y with respect to x to find the slope.

Step-by-step explanation:

To find the slope of the tangent line to the curve, we need to differentiate the given equation implicitly. Let's differentiate both sides of the equation using the chain rule.

The derivative of √(8x) is (1/2)(8x)^(-1/2) * 8 = 4√(2/x).

The derivative of -7y√(xy) is -7x√(y) - 7y(1/2)(xy)^(-1/2) * x = -7x√(y) - (7/2)y√(x).

Setting the derivative equal to zero, we have 4√(2/x) - 7x√(y) - (7/2)y√(x) = 0.

Next, we need to solve this equation for y and x in terms of each other to find the coordinates of the point where the tangent line intersects the curve.

Finally, plug the coordinates into the derivative of y with respect to x to find the slope of the tangent line.