# Relationship of two derivatives

Problem:

(a) Find the directional derivative of $w=x^2+y^2$ in the direction of the tangent vector to the spiral $(x, y) = (e^t \cos(t), 2e^t \sin(t))$, at the point defined by $t=0$. Done.

(b) Find $\frac{dw}{dt}$ along the spiral, at the same point. Done

(c) How are these rates of change related? This is the difficult part

Attempt at solution for (c):

The rate of change in (a) is the directional rate of change of $w$ in the direction of $(1, 2)$, while the rate of change in (b) is the rate of change of $w$ with respect to $t$. Both derivatives are at the same point and in the same direction, but the derivative in (b) changes faster because of the rate of change in $t$.

I feel that my solution lacks some insight. Would appreciate a hint.

• Thanks. But in part (a) we find that the directional derivative is $2/sqrt{5}$, while in part (b) it's 2. They are not the same, so it's still a question how they might be related. Calculations for part (a): (i) first I found the tangent vector to the spiral and set t to 0. (ii) I normalized the tangent vector, which came to $(1/sqrt{5}, 2/sqrt{5})$. (iii) the dot product of the gradient with the above vector came to $2/sqrt{5}$. – sequence Feb 13 '15 at 5:39