Question

Use implicit differentiation to find the slope of the tangent line to the curve y = x - 2y = x³/9 at the point (1, 10/21).

Answer 1

The student was asked to find the slope of the tangent line using implicit differentiation. The slope was found by differentiating the assumed correct equation y = (x³)/9 with respect to x, yielding dy/dx = 1/3 at the point (1, 10/21).

Step-by-step explanation:

The student asked to use implicit differentiation to find the slope of the tangent line to the curve at a specific point. The first step is to rewrite the original equation in a correct form since there seems to be a typo in the question. Assuming the correct equation is y = (x³)/9, we start by differentiating both sides with respect to x, applying the power rule to the right side:

dy/dx = (3x²)/9

To find the slope at the point (1, 10/21), we substitute x = 1 into the derived formula:

dy/dx at x = 1 = (3(1)²)/9 = 1/3

Therefore, the slope of the tangent line at the point (1, 10/21) is 1/3.