Question
Use the Chain Rule to find dw/dt.
w = ln(x^2 + y^2 + z^2), x = 2 sin(t), y = 8 cos(t), z = 5 tan(t)
w = ln(x^2 + y^2 + z^2), x = 2 sin(t), y = 8 cos(t), z = 5 tan(t)
Answers
mathhelper
from x = 2 sin(t), y = 8 cos(t), z = 5 tan(t)
dx/dt = 2cos(t)
dy/dt = -8sin(t)
dz/dt = 5sec^2 (t)
w = ln(x^2 + y^2 + z^2)
dw/dt = 1/(x^2 + y^2 + z^2) * (2x dx/dt + 2y dy/dt + 2z dz/dt)
= (4x cos(t) - 16y sin(t) + 10z sec^2 t) / (x^2 + y^2 + z^2)
= ....
replace the x, y, and z with the original expressions
dx/dt = 2cos(t)
dy/dt = -8sin(t)
dz/dt = 5sec^2 (t)
w = ln(x^2 + y^2 + z^2)
dw/dt = 1/(x^2 + y^2 + z^2) * (2x dx/dt + 2y dy/dt + 2z dz/dt)
= (4x cos(t) - 16y sin(t) + 10z sec^2 t) / (x^2 + y^2 + z^2)
= ....
replace the x, y, and z with the original expressions