Implicit differentiation practice problems
dy/dx when y is tangled up with x in an equation. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
The curve \( \displaystyle x^{3} + 2 x y + y^{3} = -2 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 3 x^{2} + 3 x y + 3 y^{2} = 3 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} y - 3 x + y^{3} = 1 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + 3 y + \sin{\left(x y \right)} = 3 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 3 x^{2} - 2 x y + 2 y^{2} = 3 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + 3 y + \sin{\left(x y \right)} = -2 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - x + y + \sin{\left(x y \right)} = 2 - \sin{\left(1 \right)} \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{3} - 2 x y + y^{3} = 5 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} y - 2 x + 3 y^{3} = -2 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{3} - 2 x y + y^{3} = 0 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 2 x^{2} - 3 x y + 2 y^{2} = 4 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 4 x + y + \sin{\left(x y \right)} = 1 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 3 x^{2} - x y + 3 y^{2} = 3 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 3 x^{2} + 2 x y + y^{2} = 1 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{3} - 2 x y + y^{3} = 2 \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{3} - 2 x y + y^{3} = 2 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 2 x^{2} + 3 x y + 2 y^{2} = 16 \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + 2 y + \sin{\left(x y \right)} = -4 - \sin{\left(1 \right)} \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 4 x^{2} + 3 x y + 2 y^{2} = 3 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} y - 4 x + 2 y^{3} = -1 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} y - x + y^{3} = -3 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 4 x^{2} - 2 x y + 3 y^{2} = 3 \) passes through \( \displaystyle (0, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{3} + 2 x y + y^{3} = 1 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle 2 x^{2} + 3 x y + 4 y^{2} = 2 \) passes through \( \displaystyle (1, 0) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} - x y + 3 y^{2} = 5 \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} y - x + 3 y^{3} = 5 \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle x^{2} - 2 x y + 3 y^{2} = 6 \) passes through \( \displaystyle (1, -1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + y + \sin{\left(x y \right)} = 3 - \sin{\left(1 \right)} \) passes through \( \displaystyle (-1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + 4 y + \sin{\left(x y \right)} = \sin{\left(1 \right)} + 2 \) passes through \( \displaystyle (1, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.
The curve \( \displaystyle - 2 x + y + \sin{\left(x y \right)} = -3 + \sin{\left(2 \right)} \) passes through \( \displaystyle (2, 1) \). Find \( \displaystyle \dfrac{dy}{dx} \) by implicit differentiation, and its value at that point.