Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable.
$$w = f(r,s,t),$$
where
$$r = r(x,y), s = s(x,y), t = t(x,y)$$
then
$$w$$
$$r,s,t$$
$$xy,xy,xy$$
Find the directional derivative of the function at the given point in the direction of the vector v.
$$g(p, q) = p^4 − p^2q^3$$
$$(1, 1)$$
$$v = i + 6j$$
$$\frac{∂g}{∂p} = 4p^3-2pq^3$$
$$\frac{∂g}{∂q} = -3p^2q^2$$
$$\vec U = \frac{1}{ \sqrt{1^2+6^2}} = \frac{1}{ \sqrt{37}}$$
$$\vec U = \frac{1}{ \sqrt{37}}\lt i+6j\gt$$
$$∇g(p,q) = (4p^3-2pq^3) + (-3p^2q^2)$$
$$∇g(1,1) = (4(1)^3-2(1)(1)^3) + (-3(1)^2(1)^2)$$
$$Dug = ∇g(1,1) \cdot \vec U$$
$$\lt 2i-3j\gt \cdot \frac{1}{ \sqrt{37}} \lt i+6j \gt$$
$$ = \frac{-16}{ \sqrt{37}}$$