Differential calculus in several variables

Compute the gradient of

$$f(x,y)=x^2{y}^3+3x-y$$

at the point $(1,1)$.

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Compute the gradient of

$$f(x,y)=x{e}^y+y^2$$

at the point $(2,0)$.

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Compute the partial derivatives of

$$f(x,y)=\sin \left(x^{2}y\right).$$

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Determine whether the function

$$f(x,y)=\{\begin{array}{ll}\frac{x^3}{x^2+y^2}& (x,y)\ne (0,0)\\ 0& (x,y)=(0,0)\end{array}$$

is differentiable at $(0,0)$.

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Determine whether the function

$$f(x,y)=x|y|$$

is differentiable at $(0,0)$.

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Compute the directional derivative of

$$f(x,y)=x^2+xy$$

at the point $(2,1)$ along the unit vector $\mathbf{v}=\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$.

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Determine the direction of steepest ascent of

$$f(x,y)=x^{2}y$$

at the point $(1,2)$ and the maximum value of the directional derivative at that point.

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Write the equation of the tangent plane to the graph of

$$f(x,y)=x^2-2y^2$$

at the point with coordinates $(2,1)$.

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Determine the point of the graph of

$$f(x,y)=x^2+y^2-4x+6y$$

at which the tangent plane is horizontal, and write the equation of the plane.

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Given $F(t)=f(x(t),y(t))$ with

$$f(x,y)=x^2+y^2,x(t)=e^t,y(t)=t^2,$$

compute $F^{\prime }(0)$.

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Let $F(s,t)=f(st,s-t)$, with $f=f(u,v)$ differentiable and

$$\nabla f(2,1)=(3,-2).$$

Compute $\frac{\partial F}{\partial s}$ and $\frac{\partial F}{\partial t}$ at the point $(s,t)=(2,1)$.

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Compute the Hessian matrix of

$$f(x,y)=x^4+y^4-4xy$$

at the point $(1,1)$.

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Compute the mixed second derivative $f_{xy}$ of

$$f(x,y)=x^3{e}^{2y}$$

and verify that it coincides with $f_{yx}$.

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Write the second-order Taylor polynomial of

$$f(x,y)=e^x\ln (1+y)$$

centered at $(0,0)$.

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