Question

For the following functions, calculate the derivative d /d t (f ◦ σ) using the chain rule

(a) f (x, y) = (x 2 + y 2 )lnx80√x 2 + y 2 ; σ(t) = (e t , e −t )

(b) f (x, y) = x ex 2+y 2 ; σ(t) = (t,−t)

(c) f (x, y, z) = x + y 2 + z 3 ; σ(t) = (cos t, sin t, t)

(d) f (x, y, z) = e x−z ( y 2 − x 2 ); σ(t) = (t, e t , t 2 )

Answer 1

We will write σ = (σ1 ,σ2 ) for the components of the curve σ. Thus if σ(t) = (e t , e −t ), then σ1 (t) = e t and σ2 (t) = e −t

(a) The partial derivatives of f are

(b) The partial derivatives of f are

Thus

d /d t (f ◦ σ) = 1 ·(−sin t) + 2 sin t · cos t + 3t 2 · 1 = −sin t + 2 sin t cos t + 3t 2

(d) The partial derivatives of f are