Q:Find the derivative of sin open parentheses square root of x close parentheses using first principles.

Asked by Ravi consul | 15th Mar, 2015, 07:50: PM

Expert Answer:

begin mathsize 14px style f left parenthesis x right parenthesis equals sin open parentheses square root of x close parentheses f left parenthesis x plus h right parenthesis equals sin open parentheses square root of x plus h end root close parentheses  fraction numerator d over denominator d x end fraction open square brackets f open parentheses x close parentheses close square 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 rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals limit as h rightwards arrow 0 of fraction numerator sin open parentheses square root of x plus h end root close parentheses minus sin open parentheses square root of x close parentheses over denominator h end fraction rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals limit as h rightwards arrow 0 of fraction numerator 2 sin open parentheses begin display style fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction end style close parentheses cos open parentheses begin display style fraction numerator square root of x plus h end root plus square root of x over denominator 2 end fraction end style close parentheses over denominator h end fraction rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals 2 limit as h rightwards arrow 0 of fraction numerator sin open parentheses fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction close parentheses over denominator open parentheses fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction close parentheses end fraction limit as h rightwards arrow 0 of fraction numerator open parentheses fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction close parentheses open parentheses fraction numerator square root of x plus h end root plus square root of x over denominator 2 end fraction close parentheses over denominator open parentheses fraction numerator square root of x plus h end root plus square root of x over denominator 2 end fraction close parentheses h end fraction limit as h rightwards arrow 0 of cos open parentheses fraction numerator square root of x plus h end root plus square root of x over denominator 2 end fraction close parentheses rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals 2 limit as h rightwards arrow 0 of fraction numerator sin open parentheses fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction close parentheses over denominator open parentheses fraction numerator square root of x plus h end root minus square root of x over denominator 2 end fraction close parentheses end fraction limit as h rightwards arrow 0 of 1 half fraction numerator x plus h minus x over denominator open parentheses square root of x plus h end root plus square root of x close parentheses h end fraction limit as h rightwards arrow 0 of cos open parentheses fraction numerator square root of x plus h end root plus square root of x over denominator 2 end fraction close parentheses rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals 2 open parentheses 1 close parentheses cross times 1 half fraction numerator 1 over denominator open parentheses 2 square root of x close parentheses end fraction cross times cos open parentheses square root of x close parentheses rightwards double arrow fraction numerator d over denominator d x end fraction open square brackets sin open parentheses square root of x close parentheses close square brackets equals fraction numerator cos open parentheses square root of x close parentheses over denominator open parentheses 2 square root of x close parentheses end fraction end style

Answered by Vijaykumar Wani | 16th Mar, 2015, 11:45: AM

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