Implicit differentiation with partial derivatives practice problems

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (-1, 1) \) for the curve \( \displaystyle x^{2} + x y^{2} = 0 \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (2, 1) \) for the curve \( \displaystyle x^{2} y + y^{3} = 5 \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (-1, 2) \) for the curve \( \displaystyle x^{2} y + y^{3} = 10 \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (2, -1) \) for the curve \( \displaystyle x^{2} y + y^{3} = -5 \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (0, 1) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = \cos{\left(1 \right)} \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (-1, 2) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = -2 - \sin{\left(1 \right)} + \cos{\left(2 \right)} \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, -1) \) for the curve \( \displaystyle x y + \sin{\left(x \right)} + \cos{\left(y \right)} = -1 + \cos{\left(1 \right)} + \sin{\left(1 \right)} \).

The equation \( \displaystyle x^{2} + x z + y^{2} z = -2 \) defines \( \displaystyle z \) as a function of \( \displaystyle x \) and \( \displaystyle y \) near \( \displaystyle (2, 2, -1) \). Find \( \displaystyle \frac{\partial z}{\partial x} \) there.

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 2) \) for the curve \( \displaystyle y^{2} + e^{x y} = 4 + e^{2} \).

Use partial derivatives to find \( \displaystyle \frac{dy}{dx} \) at \( \displaystyle (1, 1) \) for the curve \( \displaystyle y^{2} + e^{x y} = 1 + e \).