Answer:dy/dt(x=3)=54
Step-by-step explanation:
We know that we can write:
[tex]\frac{dy}{dt}=\frac{dy}{dx} *\frac{dx}{dt}[/tex]
and so evaluate it as a product of functions, that is for a given value of x or t, we get that
[tex]\frac{dy}{dt}(s)=\frac{dy}{dx} (s) * \frac{dx}{dt}(s)[/tex].
Now we are told that:
[tex]1. x=x(t),\ 2.y=x^3+4,\ \frac{dx}{dt} =2,\ for \ x=3,[/tex]
whenever it happens, this means the influence of t is hidden, and we only consider the result, that is, how much is x when the derivative has the value of 2, we are not concerned for what value of t it happens, since we get all the necessary information beforehand.
Now we need to calculate
[tex]\frac{dy}{dx}=3x^2 \rightarrow \frac{dy}{dx}(3)=3(3)^2=27[/tex].
Therefore
[tex]\frac{dy}{dt}(x=3)=\frac{dy}{dx} (x=3) * \frac{dx}{dt}(x=3)[/tex] or
[tex]\frac{dy}{dt}(x=3)=27 *2 = 54[/tex].
