91Ó°ÊÓ

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y)=x^{3} y-y^{3} $$

In Problems 1-6, find dw/dt by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=x^{2} y-y^{2} x ; x=\cos t, y=\sin t $$

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\left(4 x-y^{2}\right)^{3 / 2}\)

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ 8 x^{2}+y^{2}+8 z^{2}=16 ;(1,2, \sqrt{2} / 2) $$

Let \(f(x, y)=y / x+x y\). Find each value. (a) \(f(1,2)\) (b) \(f\left(\frac{1}{4}, 4\right)\) (c) \(f\left(4, \frac{1}{4}\right)\) (d) \(f(a, a)\) (e) \(f\left(1 / x, x^{2}\right)\) (f) \(f(0,0)\) What is the natural domain for this function?

In Problems 1-8, find the directional derivative of \(f\) at the point \(\mathbf{p}\) in the direction of \(\mathbf{a}\). \(f(x, y)=y^{2} \ln x ; \mathbf{p}=(1,4) ; \mathbf{a}=\mathbf{i}-\mathbf{j}\)

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ x^{2}-y^{2}+z^{2}+1=0 ;(1,3, \sqrt{7}) $$

Find the maximum of \(f(x, y)=4 x^{2}-4 x y+y^{2}\) subject to the constraint \(x^{2}+y^{2}=1\).

In Problems 1-16, find all first partial derivatives of each function. \(f(x, y)=\frac{x^{2}-y^{2}}{x y}\)

In Problems 1-6, find dw/dt by using the Chain Rule. Express your final answer in terms of \(t\). $$ w=e^{x} \sin y+e^{y} \sin x ; x=3 t, y=2 t $$