Let ,#f(x)=color(red)(sin)(x+1)#

#f^'(x)=color(red)(cos)(x+1)d/(dx)color(blue)((x+1)#

#f'(x)=cos(x+1)*color(blue)(1#

#:. f'(x)=cos(x+1)#

To apply the chain rule to this problem, the derivative of the outside function is multiplied by the derivative of the inside function. The outside function is the #sin# function, and the inside function is the #(x+1)#.

The derivative of #color(red)(sintheta# is #color(red)(costheta# and the derivative of #color(blue)((x+1)# is just #color(blue)(1#. This is shown in the second step. Remember that when the chain rule is applied to the outside function, the inside function remains the same, hence why it is #cos(x+1)# instead of #cosx#.

Hope this helps.