Answer: 52
Explanation:
Let [tex]D_qf (p)[/tex] = directional derivative of f at point p in the direction of q.
Then, the directional derivative is given by
[tex]D_q f(p) = \bigtriangledown f(p) \cdot q[/tex]
Note that
[tex]\bigtriangledown f(x, y ,z) = (y^2 z^3, 2xyz^3, 3xy^2 z^2)
\\ \Rightarrow \bigtriangledown f(4, 1, 1) = ((1)^2 (1)^3, 2(4)(1)(1)^3, 3(4)(1)^2 (1)^2)
\\ \Rightarrow \boxed{\bigtriangledown f(4, 1, 1) = (1, 8, 12)}[/tex]
So,
[tex]D_q f(p) = \bigtriangledown f(p) \cdot q
\\ \indent D_q f((4, 1, 1)) = \bigtriangledown f((4, 1, 1)) \cdot (0, -7, 9)
\\ \indent D_q f((4, 1, 1)) = (1, 8, 12) \cdot (0, -7, 9)
\\ \indent D_q f((4, 1, 1)) = 1(0) + 8(-7) +12(9)
\\ \indent \boxed{D_q f((4, 1, 1)) = 52}[/tex]
