AP Calculus AB Practice Quiz: Implicit Differentiation

Correct Answer: A

To find dy/dx, differentiate both sides of the equation with respect to x. This gives d/dx(x²) + d/dx(y²) = d/dx(25). Applying the power rule and the chain rule, we get 2x + 2y * (dy/dx) = 0. The dy/dx term appears because the chain rule is the basis for implicit differentiation, where y is treated as a function of x. Solving for dy/dx gives 2y(dy/dx) = -2x, which simplifies to dy/dx = -x/y.

Correct Answer: C

Implicit differentiation treats y as a function of x, i.e., y = f(x). Therefore, when differentiating a function of y with respect to x, the chain rule must be applied. For a term like y³, the outer function is (something)³ and the inner function is y. The derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This results in 3y² * (dy/dx). This demonstrates that the chain rule is the basis for implicit differentiation.

Correct Answer: A

Differentiate both sides with respect to x. The term x²y requires the product rule: d/dx(x²y) = (2x)(y) + (x²)(dy/dx). The derivative of y is dy/dx. The derivative of the constant 3 is 0. The equation becomes 2xy + x²(dy/dx) + dy/dx = 0. To solve for dy/dx, factor it out: (x² + 1)dy/dx = -2xy. Finally, divide to isolate dy/dx: dy/dx = -2xy / (x² + 1).

Correct Answer: C

The slope of the tangent line is the value of the derivative dy/dx at the given point. First, find dy/dx by implicitly differentiating the equation. d/dx(x³ + y³) = d/dx(2xy). This gives 3x² + 3y²(dy/dx) = 2y + 2x(dy/dx). The dy/dx terms arise from the chain rule. Rearrange to solve for dy/dx: 3y²(dy/dx) - 2x(dy/dx) = 2y - 3x², which gives (3y² - 2x)dy/dx = 2y - 3x². So, dy/dx = (2y - 3x²) / (3y² - 2x). Substitute the point (x, y) = (1, 1) into this expression: dy/dx = (2(1) - 3(1)²) / (3(1)² - 2(1)) = (2 - 3) / (3 - 2) = -1.

Correct Answer: A

We calculate the derivative of the implicitly defined function by differentiating both sides with respect to x. d/dx(sin(x)) + d/dx(cos(y)) = d/dx(1). The derivative of sin(x) is cos(x). For cos(y), we use the chain rule because y is a function of x: d/dx(cos(y)) = -sin(y) * (dy/dx). The derivative of 1 is 0. The equation becomes cos(x) - sin(y)(dy/dx) = 0. Solving for dy/dx, we get sin(y)(dy/dx) = cos(x), so dy/dx = cos(x) / sin(y).

Correct Answer: A

First, calculate the first derivative. Implicitly differentiating x² + y² = 9 gives 2x + 2y(dy/dx) = 0, so dy/dx = -x/y. To find the second derivative, differentiate dy/dx with respect to x using the quotient rule: d²y/dx² = -[ (1*y - x*(dy/dx)) / y² ]. The chain rule is the basis for the dy/dx term. Substitute the expression for dy/dx back into this equation: d²y/dx² = -[ (y - x*(-x/y)) / y² ] = -[ (y + x²/y) / y² ]. To simplify, multiply the numerator by y/y: d²y/dx² = -[ (y² + x²) / y ] / y² = -(y² + x²) / y³. From the original equation, we know x² + y² = 9. Substitute this back in to get d²y/dx² = -9/y³.

Correct Answer: A

To calculate the derivative of this implicitly defined function, differentiate both sides with respect to x. d/dx(e^y) = d/dx(x + y). The left side requires the chain rule: d/dx(e^y) = e^y * (dy/dx). The right side is d/dx(x) + d/dx(y) = 1 + dy/dx. The equation becomes e^y(dy/dx) = 1 + dy/dx. To solve for dy/dx, gather all dy/dx terms on one side: e^y(dy/dx) - dy/dx = 1. Factor out dy/dx: (e^y - 1)dy/dx = 1. Finally, dy/dx = 1 / (e^y - 1).