Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

y = √x

(a) Find dy/dt, given x=1 and dx/dt = 3.

dy/dt = ?

(b) Find dx/dt, given x=36 and dy/dt = 4.

dx/dt = ?

I've tried this many times, many ways but cant get my head around it. Some guidance will be greatly appreciated!

Thank you in advance.

Answer 1

Solution:

You need to realize that dy/dx can be written as (dy/dt) (dt/dx)

a) (dy/dx) = (dy/dt) (dt/dx) , y =√x , y= x ^1/2 , dy/dx = 1/2 x^1/2-1 = 1/2 x^-1/2 = 1/2( 1/ x^1/2) = 1/2 ( 1/√x)

= 1 / 2√x

find dy/dt , 1 / 2√x = (dy/dt ) (dt/dx), but dx/dt = 3 , then dt /dx = 1/3 substitute

(1 / 2√x) = ( dy/dt ) ( 1/3) , and dy/dt = (3 / 2√x) for x =1, dy/dt = 3/2

b) (dy/dx) = (dy/dt) (dt/dx) , (1 / 2√x) = (4) ( dt/dx) , dt/dx = 1 / 8 √x , and dx /dt = 8√x for x = 36

dx/dt = 8 √36 = (8)( 6) = 48