Question

Find the rate of change of f(x,y,z) = xyz in the direction normal to the surface yx^2 + xy^2 + yz^2 = 75 at (1,5,3).

I keep getting this wrong and I don't know what I did right. The answer's 147/sqrt(101) but I got 357/sqrt(1361).

Is my method right? Or something else incorrect?
Steps I did:

Gradients:
▽f(x,y,z)=(yz, xz, xy)
▽f(1,5,3)=(15,3,5)

(surface F)
▽F(x,y,z)=(2xy, x^2 + 2xy + z^2, 2yz)
▽F(1,5,3)=(10,19,30)

and then got the direction vector by doing:
▽F / ||▽F|| = (10,19,30) / sqrt(10^2 + 19^2+ 30^2) = (10,19,30) / sqrt(1361)

and then got ROC by ▽f ∙ (▽F / ||▽F||) by dot product.
= 357 / sqrt(1361)

Answer 1

well, there's this:
F(x,y,z) = yx^2 + xy^2 + yz^2
▽F(x,y,z)=(2xy+y^2, x^2 + 2xy + z^2, 2yz)
▽F(1,5,3)=(35,20,30)
so |▽F| = 5√101