Define quotient rule, product rule, and chain rule .

Asked by NAKSHATRA | 1st Dec, 2014, 09:24: AM

Expert Answer:

Q u o t i e n t space r u l e : L e t space f open parentheses x close parentheses space a n d space g open parentheses x close parentheses space b e space t w o space f u n c t i o n s. L e t space h open parentheses x close parentheses equals fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction G i v e n space y equals h open parentheses x close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator d h over denominator d x end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction open square brackets fraction numerator f open parentheses x close parentheses over denominator g open parentheses x close parentheses end fraction close square brackets rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator g open parentheses x close parentheses cross times begin display style fraction numerator d f open parentheses x close parentheses over denominator d x end fraction end style minus f open parentheses x close parentheses cross times fraction numerator d g open parentheses x close parentheses over denominator d x end fraction over denominator g open parentheses x close parentheses squared end fraction  P r o d u c t space r u l e : L e t space f open parentheses x close parentheses space a n d space g open parentheses x close parentheses space b e space t w o space f u n c t i o n s. L e t space h open parentheses x close parentheses equals f open parentheses x close parentheses cross times g open parentheses x close parentheses G i v e n space y equals h open parentheses x close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator d h over denominator d x end fraction rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction open square brackets f open parentheses x close parentheses cross times g open parentheses x close parentheses close square brackets rightwards double arrow fraction numerator d y over denominator d x end fraction equals f open parentheses x close parentheses cross times fraction numerator d g open parentheses x close parentheses over denominator d x end fraction plus g open parentheses x close parentheses cross times fraction numerator d f open parentheses x close parentheses over denominator d x end fraction  C h a i n space r u l e : L e t space f open parentheses x close parentheses space a n d space g open parentheses x close parentheses space b e space t w o space f u n c t i o n s. L e t space y open parentheses x close parentheses equals open parentheses f ring operator g close parentheses open parentheses x close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction open parentheses f ring operator g close parentheses open parentheses x close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals fraction numerator d over denominator d x end fraction open parentheses f open parentheses g open parentheses x close parentheses close parentheses close parentheses rightwards double arrow fraction numerator d y over denominator d x end fraction equals f open parentheses g open parentheses x close parentheses close parentheses apostrophe rightwards double arrow fraction numerator d y over denominator d x end fraction equals f apostrophe open parentheses g open parentheses x close parentheses close parentheses cross times g apostrophe open parentheses x close parentheses

Answered by Vimala Ramamurthy | 2nd Dec, 2014, 10:41: AM