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Evaluate $space space limit as straight x rightwards arrow straight x over 4 of fraction numerator cos space straight x space minus space sin space straight x over denominator straight x minus straight pi divided by 4 end fraction.$

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22.

Evaluate $space space space space space limit as straight x rightwards arrow straight pi of fraction numerator sin left parenthesis straight n minus straight pi right parenthesis over denominator straight pi left parenthesis straight pi minus straight x right parenthesis end fraction.$

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23.

Evaluate $space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator sin space space ax plus bx over denominator ax plus sin space bx end fraction comma space straight a comma space space straight b comma space straight a plus straight b not equal to 0$

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24.

Evaluate $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space limit as straight x rightwards arrow 0 of left parenthesis cosec space straight x minus cot space straight x right parenthesis$

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25. Find the derivative of x²-2 at x = 10.

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26. Find the derivative of f (x) = 7 at x = 3

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27.

Find the derivative of $space fraction numerator 1 over denominator square root of 3 straight x plus 1 end root end fraction$ at x =2.

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28. Find the derivative of the following functions from first principle.

(x-1) (x-2)

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29. Find the derivative of the following functions from first principle.

$space space space space space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction$

space space space space straight f left parenthesis straight x right parenthesis space fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space therefore space space space space space space space space straight f left parenthesis straight x right parenthesis space equals space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight x plus straight h right parenthesis minus straight f left parenthesis straight x right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator begin display style fraction numerator straight x plus straight h plus 1 over denominator straight x plus straight h minus 1 end fraction minus fraction numerator straight x plus 1 over denominator straight x minus 1 end fraction end style over denominator straight h end fraction$

$space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator left parenthesis straight x plus 1 plus straight h right parenthesis left parenthesis straight x minus 1 right parenthesis minus left parenthesis straight x minus 1 plus straight h right parenthesis left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets$

$space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus left parenthesis straight x squared minus 1 right parenthesis minus straight h left parenthesis straight x plus 1 right parenthesis over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets$

$space space space space space space space space space space space space space space space space space space space space space space space space space equals space limit as straight h rightwards arrow 0 of 1 over straight h open square brackets fraction numerator straight x squared minus 1 plus hx minus straight h minus straight x squared plus 1 minus hx minus straight h over denominator left parenthesis straight x plus straight h minus 1 right parenthesis left parenthesis straight x minus 1 right parenthesis end fraction close square brackets$

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