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AI & ML · Multivariable Calculus

Gradients & directional 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)=−4x2−3y2−2xy+4x−5yf(x,y) = -4x^{2} - 3y^{2} - 2xy + 4x - 5y. Find the gradient vector ∇f\nabla f at the point (−1,3)(-1, 3): (i) the xx-component ∂f∂x\dfrac{\partial f}{\partial x}, and (ii) the yy-component ∂f∂y\dfrac{\partial f}{\partial y}.

  2. Warm-up

    Let f(x,y)=2x2+2y2+2xy+4x+2yf(x,y) = 2x^{2} + 2y^{2} + 2xy + 4x + 2y. Find the gradient vector ∇f\nabla f at the point (−2,−1)(-2, -1): (i) the xx-component ∂f∂x\dfrac{\partial f}{\partial x}, and (ii) the yy-component ∂f∂y\dfrac{\partial f}{\partial y}.

  3. Warm-up

    Let f(x,y)=−3x2−4y2−2xy+2x−2yf(x,y) = -3x^{2} - 4y^{2} - 2xy + 2x - 2y. Find the gradient vector ∇f\nabla f at the point (−1,2)(-1, 2): (i) the xx-component ∂f∂x\dfrac{\partial f}{\partial x}, and (ii) the yy-component ∂f∂y\dfrac{\partial f}{\partial y}.

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