Question

Find the derivative of the given function with respect to the independent variable x or t. The symbols a, b and c are constants greater than 1. You are not required to combine like terms, reduce fractions, or otherwise simplify your final answer.

(1) y = (2/[ a+bx])^3
(2) y = (at^3 - 3bt)^3
(3) y = (t^b)e^(b/t)
(4) z = ax^2.sin (4x)

Answer 1

(1)
`y=((2)/(a+bx))^3`

By differentiating w.r.t. x,

`(dy)/(dx)=3((2)/(a+bx))^2* (d)/(dt)((2)/(a+bx))`

`=3((2)/(a+bx))^2* (-(2)/((a+bx)^2))`

`=-(24)/((a+bx)^4)`

(2)
`y=(at^3-3bt)^3`

By differentiating w.r.t. t,

`(dy)/(dt)=3(at^3-3bt)^2* (d)/(dt)(at^3-3bt)`

`=3(at^3-3bt)^2 (3at^2-3b)`

`=9t^2(at^2-3b)^2(at^2-b)`

(3)
`y=(t^b)(e^(b)/(t))`

Differentiating w.r.t. t,

`(dy)/(dt)=t^b* (d)/(dt)(e^(b)/(t))+(d)/(dt)(t^b)* e^(b)/(t)`

`=t^b(e^(b)/(t))* (d)/(dt)((b)/(t)) + bt^(b-1)(e^(b)/(t))`

`=t^be^(b)/(t)(-(b)/(t^2))+bt^(b-1)e^{(b)/(t)}`

(4)
`z = ax^2.sin (4x)`

Differentiating w.r.t. x,

`(dz)/(dt)=ax^2* (d)/(dx)(sin (4x))+sin (4x)* (d)/(dx)(ax^2)`

`=ax^2* cos(4x).4+sin (4x)(2ax)`

`=4ax^2cos (4x)+2ax sin (4x)`