Question

How do you differentiate `f(x)=sqrtsin((3x+7)^3)` using the chain rule.?

Answer 1

Using chain rule consists in dealing with the separate operations in the equation one at a time.

Explanation:

First, chain rule states that `(dy)/(dx)=(dy)/(du)(du)/(dx)`, or, in this case, we'll use a longer one, following the same logic: `(dy)/(dx)=(dy)/(du)(du)/(dv)(dv)/(dw)(dw)/(dx)`

Here, we can rename `y=sqrt(u)`, `u=sin(v)`, `v=(w)^3` and `w=3x+7`

Following the statement of the chain rule:

`(dw)/(dx)=3`

`(dv)/(dw)=3w^2`

`(du)/(dv)=cos(v)`

`(dy)/(du)=1/(2sqrt(u))`

Aggregating them...

`(dy)/(dx)=1/(2sqrt(u))(cos(v))(3w^2)(3)`

Substituting `u`, `v` and `w`:

`(dy)/(dx)=(cos(w^3)(3(3x+7)))/(2sqrt(sin(v)))`

`(dy)/(dx)=(cos(3x+7)^2(9x+21))/(2sqrt(sin(w^3)))`

`(dy)/(dx)=(cos(3x+7)^2(9x+21))/(2sqrt(sin(3x+7)^3))`