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If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent minus 1 right parenthesis end fraction(x not equal to 0), then stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis is-

Maths-General

  1. 0    
  2. 5/3    
  3. 3/5    
  4. None    

    Answer:The correct answer is: 3/5f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis d x over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
    stack l i m with x rightwards arrow 0 below f ´ left parenthesis x right parenthesis = fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of negative 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 right parenthesis end fraction
    = fraction numerator x to the power of 1 divided by 3 end exponent fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction. fraction numerator log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction.3 x over denominator x. open parentheses fraction numerator tan to the power of negative 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator e to the power of 5 x to the power of 1 divided by 3 end exponent end exponent minus 1 over denominator 5 x to the power of 1 divided by 3 end exponent end fraction close parentheses.5 x to the power of 1 divided by 3 end exponent end fraction
    = fraction numerator 1.1.3 over denominator 1.1.5 end fraction equals 3/5

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    General
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    If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent – 1 right parenthesis end fraction (x not equal to 0), then stack l i m with x rightwards arrow 0 below f ' (x) is -

     f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 right parenthesis end fraction
    stack l i m with x rightwards arrow 0 below f'(x)
    stack l i m with x rightwards arrow 0 below fraction numerator open parentheses fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses x to the power of 1 divided by 3 end exponent open square brackets fraction numerator cos invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction close square brackets left parenthesis 3 x right parenthesis over denominator open parentheses fraction numerator tan to the power of – 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent. x. open parentheses fraction numerator e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses. x to the power of 1 divided by 3 end exponent end fraction
    = 3

    If f(x) is the primitive of fraction numerator sin invisible function application root index 3 of x end root log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of root index 3 of x end root end exponent – 1 right parenthesis end fraction (x not equal to 0), then stack l i m with x rightwards arrow 0 below f ' (x) is -

    maths-General
     f(x) = not stretchy integral fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent log invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator left parenthesis tan to the power of – 1 end exponent invisible function application square root of x right parenthesis to the power of 2 end exponent left parenthesis e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 right parenthesis end fraction
    stack l i m with x rightwards arrow 0 below f'(x)
    stack l i m with x rightwards arrow 0 below fraction numerator open parentheses fraction numerator sin invisible function application x to the power of 1 divided by 3 end exponent over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses x to the power of 1 divided by 3 end exponent open square brackets fraction numerator cos invisible function application left parenthesis 1 plus 3 x right parenthesis over denominator 3 x end fraction close square brackets left parenthesis 3 x right parenthesis over denominator open parentheses fraction numerator tan to the power of – 1 end exponent invisible function application square root of x over denominator square root of x end fraction close parentheses to the power of 2 end exponent. x. open parentheses fraction numerator e to the power of x to the power of 1 divided by 3 end exponent end exponent – 1 over denominator x to the power of 1 divided by 3 end exponent end fraction close parentheses. x to the power of 1 divided by 3 end exponent end fraction
    = 3
    General
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    log subscript x subscript 1 end subscript end subscript invisible function application blanklog subscript x subscript 3 end subscript end subscript invisible function application blank...log subscript x subscript n minus 1 end subscript end subscript invisible function application blank open parentheses x subscript n minus 1 end subscript to the power of x subscript n minus 2 end subscript superscript. to the power of. to the power of. x subscript 1 end subscript end exponent end exponent end superscript end exponent log subscript x subscript n end subscript end subscript invisible function application x subscript n end subscript close parentheses
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    maths-

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