Using the chain rule, differentiate the function f(x)=square root(5+16x-(4x)squared). what is the derivative of the square root?

As per chain rule of
differentiation:

dy/dx =
(dy/du)(du/dv)(dv/dx)

Where:

y
= f(u)

u = f(v) and

v =
f(x)

To differentiate the given function using this rule we
proceed as follows.

Given:

y =
[5 + 16x - (4x)^2]^1/2

Let:

u
= 5 + 16x - (4x)^2

v =
4x

Then:

y = f(u) = u^1/2
and

u = f(v) = 5 + 4v -
v^2

dy/du = (1/2)u^(-1/2)

=
(1/2)(5 + 4v - v^2)^(-1/2)

= (1/2)[5 + 16x -
(4x)^2]^(-1/2)

= (1/2)[5 + 16x -
16x^2]^(-1/2)

du/dv = 4 - 2v

=
4 - 8x

dv/dx = 4

Then we
calculate derivative of given expression as:

dy/dx =
(dy/du)(du/dv)(dv/dx)

= {(1/2)[5 + 16x - (4x)^2]^(-1/2)}(4
- 8x)4

= 16(1 - 2x){(1/2)[5 + 16x -
(4x)^2]^(-1/2)}

= 16(1 - 2x){(1/2)[5 + 16x -
(4x)^2]^(-1/2)}

= 8(1 - 2x)(5 + 16x -
16x^2)^(-1/2)