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

1 Answer
Feb 18, 2016

Answer:

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)#