Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. xy = 4 (a) Find dy/dt, given x = 8 and dx/dt = 15. dy/dt = (b) Find dx/dt, given x = 1 and dy/dt = –5. dx/dt =

Answer 1

Answer:

a) -0.9375

b) 1.25

Step-by-step explanation:

We are given the following in the question:

$f(x,y) = 4$

where x and y are both differentiable functions of t.

a)  x = 8 and dx/dt = 15

$\\xy = 4\\\\y = \dfrac{4}{x}\\\\y = \dfrac{4}{8}=\dfrac{1}{2}\\\\\dfrac{d(f(x,y))}{dt} = 0\\\\x\dfrac{dy}{dt} + y\dfrac{dx}{dt} = 0\\\\8\dfrac{dy}{dt} + \dfrac{1}{2}(15) = 0 \\\\\dfrac{dy}{dt} = \dfrac{1}{8}\times \dfrac{-15}{2}\\\\\dfrac{dy}{dt} = -\dfrac{15}{16}\\\\\dfrac{dy}{dt}=-0.9375$

b) x = 1 and dy/dt = –5

$xy = 4\\\\y = \dfrac{4}{x}\\\\ y= 4\\\\\dfrac{d(f(x,y))}{dt} = 0\\\\x\dfrac{dy}{dt} + y\dfrac{dx}{dt} = 0\\\\(1)(-5) + 4\dfrac{dx}{dt} = 0 \\\\\dfrac{dx}{dt} = \dfrac{5}{4}\\\\\dfrac{dx}{dt} = 1.25$