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Question

# Find the derivative of the following functions from first principle: (i) -x (ii) (−x)−1 (iii) sin (x+1) (iv) cos(x−π8)

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Solution

## (i) Here f(x)= -x Then f(x+h)= -(x+h) We know that: f′(x)=limh→0f(x+h)−f(x)h ⇒f′(x)=limh→0−(x+h)−(−x)h =limh→0−x−h+xh =limh→0−hh=−1 (ii) Here f(x)= (−x)−1=−1x Then f(x+h)=−1x+h We know that: f′(x)=limh→0−1x+h−f(x)h =limh→0−x+x+hhx(x+h) =limh→0hhx(x+h)=1x2 (iii) Here f(x)= sin (x+1) Then f(x+h)= sin (x+h+1) We know that f′(x)=limh→0f(x+h)−f(x)h ⇒f′(x)=limh→0sin(x+h+1)−sin(x+1)h =limh→02cos (2x+h+22) sin (h2)h =limh→0cos(x+1+h2) sin(h2)(h2) =cos (x+1) (iv) Here f(x)=cos(x−π8) Then f(x+h)=cos(x+h−π8) We know that f′(x)=limh→0f(x+h)−f(x)h ⇒f′(x)=limh→0cos(x+h−π8)−cos(x−π8)h =limh→0−2sin(x−π8+h2) sin(h2)h =limh→0−sin(x−π8+h2).sin(h2)(h2) =−sin(x−π8).

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