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

Explanation:

We know that we can write:

`(dy)/(dt)=(dy)/(dx) *(dx)/(dt)`

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

`(dy)/(dt)(s)=(dy)/(dx) (s) * (dx)/(dt)(s)`.

Now we are told that:

`1. x=x(t),\ 2.y=x^3+4,\ (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

`(dy)/(dx)=3x^2 \rightarrow (dy)/(dx)(3)=3(3)^2=27`.

Therefore

`(dy)/(dt)(x=3)=(dy)/(dx) (x=3) * (dx)/(dt)(x=3)` or

`(dy)/(dt)(x=3)=27 *2 = 54`.