Select Page

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}} \$\$