directional derivative with increasing path

Question

$f(x,y,z) = xe^{y^2-z^2}$
$P(1,2,-2)$
$r(t)=(t)i+(2cos(t-1))j-(2e^{t-1})k$

The directional derivative goes along the path r(t) (I assume this means parallel and r(t) is increasing)

For this problem, I have calculated the gradient but how do I perform the dot product if $r$ has the variable $t$? Should I leave it as just the multiplication of the terms in the gradient and path?

Answer 1

You can express the gradient of f in terms of t, using the fact that the directional derivative is computed along points in the path r. Substitute x(t), y(t), z(t) in $\nabla f$.