Question

Find dy du , du dx , and dy dx .

(a) y = u5 and u = x2 + 1 dy du = du dx = dy dx =

(b) y = u4 and u = 4x2 − x + 6 dy du = du dx = dy dx =

Answer 1

have fun with the answer just want some points

Explanation:

Answer 2

a)
`(dy)/(du)=5(x^2 +1)^4`,
`(du)/(dx)=2x`,
`(dy)/(dx)=10x(x^2+1)^4`

b)
`(dy)/(du)=4(4x^2-x+6)^3`,
`(du)/(dx)=8x-1`,
`(dy)/(dx)=4(8x-1)(4x^2-2+6)^3`

Explanation:

We can use the chain rule in the following form: is u=u(x) is a differentiable function depending on x and y=y(u) is a differentiable function depending on u, then
`(dy)/(dx)=(dy)/(du) (du)/(dx)`.

a)
`(dy)/(du)=(d)/(du) (u^5)=5u^4=5(x^2 +1)^4` from the power rule.

`(du)/(dx)=(d)/(dx) (x^2 +1)=2x`.

From the previous parts and the chain rule,
`(dy)/(dx)=(dy)/(du) (du)/(dx)=5(x^2 +1)^4(2x)=10x(x^2+1)^4`

b)
`(dy)/(du)=(d)/(du) (u^4)=4u^3=4(4x^2-x+6)^3`

`(du)/(dx)=(d)/(dx) (4x^2-x+6)=8x-1` from the power and sum rules.

Then,
`(dy)/(dx)=(dy)/(du) (du)/(dx)=4(4x^2 -x+6)^3(8x-1)=4(8x-1)(4x^2-x+6)^3`