How do you find f'(x) using the definition of a derivative for f(x)=cos⁡x?

Question

How do you find f&#039;(x) using the definition of a derivative for f(x)=cos⁡x?

annie996k88v · Accepted Answer

Explanation:According to the definition:f′(x0)=limh→0f(x0+h)−f(x0)hIf we use this definition for f(x)=cos⁡x we get:f′(x0)=limh→0cos⁡(x0+h)−cos⁡x0hf′(x0)=limh→0−2sin⁡(x0+h+x2)⋅sin⁡(x0+h−x02)hf′(x0)=limh→0−2sin⁡(x0+h2)sin⁡(h2)hf′(x0)=limh→0−sin⁡(x0+h2)sin⁡(h2)h2f′(x0)=limh→0(−sin⁡(x0+h2))⋅limh→0sin⁡(h2)h2For further calculation we will use the identity:limx→0sin⁡xx=1So the left limit is −sin⁡x0, and the right is 1. Finally we get, thatf′(x0)=−sin⁡x0

mortaraskxnl · Accepted Answer

f(x)=cos⁡xf′(x)=limh→0cos⁡(x+h)−cos⁡xh=limh→0cos⁡xcos⁡h−sin⁡xsin⁡h−cos⁡xh=limh→0cos⁡xcos⁡h−cos⁡xh−sin⁡xsin⁡hh=limh→0cos⁡xcos⁡h−cos⁡xh−sin⁡xsin⁡hh=limh→0(cos⁡xcos⁡h−1h−sin⁡xsin⁡hh)=[limh→0cos⁡x][limh→0cos⁡h−1h]−[limh→0sin⁡x][limh→0sin⁡hh]=[0][1]−[sin⁡x][1]=−sin⁡xThat is: f′(x)=−sin⁡x

How do you find f'(x) using the definition

Answered question

Answer & Explanation

New Questions in Calculus 1