For the circle x^2 + y^2 = 25, dy/dx equals -x/y. What is dy/dx at the point (3,4)?

For the circle x^2 + y^2 = 25, dy/dx equals -x/y. What is dy/dx at the point (3,4)?

• A. 3/4
• B. -3/4
• C. -3
• D. 4/3

Answer: (B)

For a circle, use implicit differentiation. Differentiate x^2 + y^2 = 25 with respect to x, treating y as a function of x: 2x + 2y dy/dx = 0. Solve for dy/dx to get dy/dx = -x/y. At the point (3,4), substitute x = 3 and y = 4: dy/dx = -3/4. So the tangent slope there is -3/4. The other values would come from dropping the negative sign, using x/y instead of -x/y, or taking a reciprocal, which aren’t correct for this derivative.

For a circle, use implicit differentiation. Differentiate x^2 + y^2 = 25 with respect to x, treating y as a function of x: 2x + 2y dy/dx = 0. Solve for dy/dx to get dy/dx = -x/y. At the point (3,4), substitute x = 3 and y = 4: dy/dx = -3/4. So the tangent slope there is -3/4. The other values would come from dropping the negative sign, using x/y instead of -x/y, or taking a reciprocal, which aren’t correct for this derivative.