Let be a non-zero real constant and consider the function

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

a

be a non-zero real constant and consider the function

p(x)=\cot(ax+b)

, where

b

is a real constant. The domain of

p

excludes any

x

where

\sin(ax+b)=0

. (a) Express

p(x)

using elementary trigonometric functions only, then differentiate to obtain

p'(x)

. (b) Show that the derivative you obtained simplifies to

p'(x)=-a\csc^{2}(ax+b)

. Clearly indicate each algebraic step in your simplification.