Question

Find the differential of each function. (a) y = tan( 5t ) dy = Correct: Your answer is correct. (b) y = 5 − v2 5 + v2

Answer 1

`(a)`
`dy = 5sec^2(5t) \ dt`

`(a) \ dy = (-20v)/((5+v^2)^2) \ dt`

Explanation:

Given;

(a) y = tan(5t)

`(b) \ y = (5-v^2)/(5+v^2)`

Solving for (a)

y = tan(5t)

let u = 5t

⇒y = tan(u)

du/dt = 5

dy/du = sec²u

`(dy)/(dt) =(dy)/(du) *(du)/(dt) \\\\(dy)/(dt) = sec^2(u)*5\\\\(dy)/(dt) = 5sec^2(u)\\\\(dy)/(dt) = 5sec^2(5t)`

`dy = 5sec^2(5t) \ dt`

Solving for b;

let u = 5 - v²

du/dv = -2v

let v = 5+ v²

dv/du = 2v

`(dy)/(dv) = (vdu - udv)/(v^2) \\\\(dy)/(dv) = (-2v(5+v^2) - 2v(5-v^2))/((5+v^2)^2)\\\\(dy)/(dv) = (-10v-2v^3-10v+2v^3)/((5+v^2)^2)\\\\`

`(dy)/(dv) = (-10v-10v)/((5+v^2)^2)\\\\(dy)/(dv) = (-20v)/((5+v^2)^2)\\\\dy = (-20v)/((5+v^2)^2) dt`