We have the formula:

$$ f'( x)= lim _ {h to 0} frac {f( x+ h)- f( x)} {h} $$

We usage the square origin feature, $latex f( x)= sqrt {x} $, to revise the numerator:

$$ f'( x)= lim _ {h to 0} frac {sqrt {x+ h} -sqrt {x}} {h} $$

The expression in the numerator can be streamlined by increasing both the numerator and also by the conjugate of the numerator:

$$ f'( x)= lim _ {h to 0} frac {(sqrt {x+ h} -sqrt {x} )( sqrt {x+ h} +sqrt {x})} {h( sqrt {x+ h} +sqrt {x})} $$

$$ f'( x)= lim _ {h to 0} frac {x+ h-x} {h( sqrt {x+ h} +sqrt {x})} $$

$$ f'( x)= lim _ {h to 0} frac {h} {h( sqrt {x+ h} +sqrt {x})} $$

Now, we can streamline the h in the numerator with the h in the :

$$ f'( x)= lim _ {h to 0} frac {1} {sqrt {x+ h} +sqrt {x}} $$

Finally, we can resolve the limitation by replacing $latex h= 0$ right into the expression:

$$ f'( x)= lim _ {h to 0} frac {1} {sqrt {x +0} +sqrt {x}} $$

$$ f'( x)= lim _ {h to 0} frac {1} {sqrt {x} +sqrt {x}} $$

$$ f'( x)= lim _ {h to 0} frac {1} {2sqrt {x}} $$



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