Question

Assume that x equals x (t )and y equals y (t ). Let y equals x cubed plus 4 and StartFraction dx Over dt EndFraction ​=2 when xequals3. Find StartFraction dy Over dt EndFraction when xequals3.

Answer 1

Answer:dy/dt(x=3)=54

Step-by-step explanation:

We know that we can write:

$\frac{dy}{dt}=\frac{dy}{dx} *\frac{dx}{dt}$

and so evaluate it as a product of functions, that is for a given value of x or t, we get that  

$\frac{dy}{dt}(s)=\frac{dy}{dx} (s) * \frac{dx}{dt}(s)$.

Now we are told that:

$1. x=x(t),\ 2.y=x^3+4,\ \frac{dx}{dt} =2,\ for \ x=3,$

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

$\frac{dy}{dx}=3x^2 \rightarrow \frac{dy}{dx}(3)=3(3)^2=27$.

Therefore

$\frac{dy}{dt}(x=3)=\frac{dy}{dx} (x=3) * \frac{dx}{dt}(x=3)$ or

$\frac{dy}{dt}(x=3)=27 *2 = 54$.