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Home / Doubts and Solutions/KERALA/Class 11 Science/Mathematics

Find the derivative of f(x)=Sin x using the first priciple.
[I got everything till the last step. How can cos(x+(h/2)).(sin h/2)/(h/2)=cosx.1?]
 
[also in short what exatly is the first principle?]
 
 
 
 
 
 
 
 
 
 
 

Asked by Deebu 1st February 2015, 1:31 AM
Answered by Expert
Answer:
T h e space p r o c e s s space o f space f i n d i n g space t h e space d e r i v a t i v e space o f space a space f u n c t i o n space b y space u sin g space t h e space d e f i n i t i o n fraction numerator d over denominator d x end fraction open curly brackets f open parentheses x close parentheses close curly brackets equals limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction i s space c a l l e d space t h e d i f f e r e n t i a t i o n space f r o m space f i r s t space p r i n c i p l e. N o w space c o n s i d e r f open parentheses x close parentheses equals sin x rightwards double arrow f open parentheses x plus h close parentheses equals sin open parentheses x plus h close parentheses S o comma space f open parentheses x plus h close parentheses minus f open parentheses x close parentheses equals sin open parentheses x plus h close parentheses minus sin x L e t space u s space u s e space t h e space t r i g o n o m e t r i c space i d e n t i t y sin C minus sin D equals 2 cos fraction numerator C plus D over denominator 2 end fraction sin fraction numerator C minus D over denominator 2 end fraction T h u s comma space sin open parentheses x plus h close parentheses minus sin equals 2 cos fraction numerator 2 x plus h over denominator 2 end fraction sin h over 2 rightwards double arrow sin open parentheses x plus h close parentheses minus sin equals 2 cos open parentheses x plus h over 2 close parentheses sin h over 2 T h u s space f r o m space t h e space d e f i n i t i o n space o f space t h e space d e r i v a t i v e comma fraction numerator d y over denominator d x end fraction equals limit as h rightwards arrow 0 of fraction numerator f open parentheses x plus h close parentheses minus f open parentheses x close parentheses over denominator h end fraction space space space space space space equals limit as h rightwards arrow 0 of fraction numerator 2 cos open parentheses x plus h over 2 close parentheses sin h over 2 over denominator h end fraction space space space space space space equals limit as h rightwards arrow 0 of fraction numerator cos open parentheses x plus h over 2 close parentheses sin h over 2 over denominator begin display style h over 2 end style end fraction space space space space space space equals limit as h rightwards arrow 0 of cos open parentheses x plus h over 2 close parentheses cross times limit as h rightwards arrow 0 of fraction numerator sin h over 2 over denominator begin display style h over 2 end style end fraction space space space space space space space space space space space equals cos x cross times limit as h rightwards arrow 0 of fraction numerator sin h over 2 over denominator begin display style h over 2 end style end fraction N o w space u s e space t h e space r e s u l t space limit as theta rightwards arrow 0 of fraction numerator sin theta over denominator begin display style theta end style end fraction equals 1 T h u s comma fraction numerator d y over denominator d x end fraction equals cos x cross times 1 rightwards double arrow space fraction numerator d y over denominator d x end fraction equals cos x
Answered by Expert 1st February 2015, 5:42 PM
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