Respuesta :

frika
If [tex]xy + x = 12 [/tex], then
[tex] \dfrac{d}{dt}(xy+x)= \dfrac{d}{dt}(12)[/tex].
Use the chain rule:
[tex] y\dfrac{dx}{dt}+x \dfrac{dy}{dt}+ \dfrac{dx}{dt}=0[/tex],
Since x=2, y=5, dx/dt = -3, then

[tex]5\cdot (-3)+2\cdot \dfrac{dy}{dt}+(-3)=0, \\ 2\cdot \dfrac{dy}{dt}=18, \\ \dfrac{dy}{dt}=9[/tex].

By using chain rule if xy + x = 12 and dx/dt = -3, then dy/dt when x = 2 and y = 5 would be 9.

What is the chain rule?

In calculus, the chain rule is defined as the formula that expresses the derivative of the composition of two functions g and f

d/dx f(g(x)) = f'(g(x)) . g'(x)

The chain rule is also expressed in Leibniz's notation.

[tex]\frac{dz}{dx} =\frac{dz}{dy }\times\frac{dy}{dx}[/tex]

It is given that xy + x = 12 and dx/dt = -3, we need to find dy/dt when x = 2 and y = 5.

xy + x = 12

on differentiating both side with respect to t

[tex]\rm \frac{d}{dt} (xy + x) = \frac{d}{dt} (12)[/tex]

by using the chain rule, we get

[tex]\rm y\frac{dx}{dt} + x\frac{dy}{dt}+\frac{dx}{dt}=0[/tex]

since dx/dt = -3, x = 2 and y = 5

[tex]\rm 5(-3)+ 2\frac{dy}{dt}+(-3)=0\\\rm 2\frac{dy}{dt} =18\\\rm \frac{dy}{dt} = 9[/tex]

Thus, by using chain rule if xy + x = 12 and dx/dt = -3, then dy/dt when x = 2 and y = 5 would be 9.

Learn more about chain rule;

https://brainly.com/question/14980705

Otras preguntas