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AI & ML · Multivariable Calculus

Partial derivatives

80 problems in this skill, each with a solver-verified answer key. Answers are checked the moment you submit — with a hint if you want one and a worked solution afterwards.

What the problems look like

A few statements from this skill. Answers, hints and worked solutions live in the practice session.

  1. Warm-up

    Let f(x,y)=−4x2y2−3xy−2x3+3y4f(x,y) = -4x^{2}y^{2} - 3xy - 2x^{3} + 3y^{4}. Find the mixed second-order partial derivative ∂2f∂x ∂y\dfrac{\partial^{2} f}{\partial x\, \partial y} at the point (1,−3)(1, -3).

  2. Warm-up

    Let f(x,y)=−2x2y2−4xy+2x3+2y4f(x,y) = -2x^{2}y^{2} - 4xy + 2x^{3} + 2y^{4}. Find the mixed second-order partial derivative ∂2f∂x ∂y\dfrac{\partial^{2} f}{\partial x\, \partial y} at the point (−1,0)(-1, 0).

  3. Warm-up

    Let f(x,y)=−4x2y2+3xy+2x3−3y4f(x,y) = -4x^{2}y^{2} + 3xy + 2x^{3} - 3y^{4}. Find the mixed second-order partial derivative ∂2f∂x ∂y\dfrac{\partial^{2} f}{\partial x\, \partial y} at the point (−2,1)(-2, 1).

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