Find the first partial derivatives of the function.

$$f(x,y)=x^4y^3+8x^2y$$

Find the directional derivative of

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

at x = 2 and y = 2 in the direction of h = 2i + 3j

Find the directional derivative of f(x, y)=x$^2$+4y$^2$ at the point P=(3,2) in the direction pointing to the origin.

Find the directional derivative of f at P in the direction of a. f(x, y)=4x³y²; P(2, 1); a=4i=3j

Find the directional derivative of the function at P in the direction of v. f(x, y) = x³ - y³, P(4, 3), v=√2/2 (i+j)

Find the directional derivative of the function at P in the direction of Q. f(x,y) = x² + 3y², P(1, 1), Q(4, 5)

Find the directional derivative of the function $f$ at the point $P$ in the direction of the vector $\mathbf{v}$.

$$f(x,y)=x^3-x^2y^2+xy+y^2;\hspace{6pt}P(1,-1),\hspace{6pt}\mathbf{v}=\mathbf{i}-2\mathbf{j}$$

Find the first partial derivatives of the function. z = tan xy

Let

$$f(x, y)= \begin{cases}\frac{x^3 y-x y^3}{x^2+y^2} & \text { if }(x, y) \neq(0,0) \\ 0 & \text { if }(x, y)=(0,0)\end{cases}$$

Does the result of the previous part contradict Clairaut's Theorem?

What does Clairaut's Theorem say?

What does Clairaut's Theorem say?

What does Clairaut's Theorem say?

What does Clairaut’s Theorem say?

What does Clairaut’s Theorem say?