Question

How do you differentiate `(x+1)^8`?

Answer 1

Use the Chain Rule to get
`color(white)("XXX")(dcolor(white)("x")(x+1)^8)/(dx) = 8(x+1)^7`

Explanation:

Chain Rule for Derivatives
`color(white)("XXX")(dcolor(white)("x")g(f(x)))/(dx) = color(red)((dcolor(white)("x")g(f(x)))/(dcolor(white)("x")f(x))) * color(blue)((dcolor(white)("x")(f(x)))/(dx))`

Let
`color(white)("XXX")f(x)=x+1`
and
`color(white)("XXX")g(x)=x^8`

Note that
`color(white)("XXX")(color(blue)((dcolor(white)("x")f(x))/(dx))=color(blue)(1))`
and
`color(white)("XXX")(dcolor(white)("x")(g(x)))/(dx)=8x^7`
`color(white)("XXXXXX")rarr color(red)( (dcolor(white)("x")g(f(x)))/(dcolor(white)("x")f(x)))=8(f(x))^7 = color(red)(8(x+1)^7)`

So
`(dcolor(white)("x")(x+1)^8)/(dx) = color(red)(8(x+1)^7)*color(blue)(1)`