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To find the derivative of $f(x)=\sqrt{x}$, substitute into the difference quotient $\frac{\sqrt{x+h}-\sqrt{x}}{h}$. Multiply the numerator and denominator by the conjugate $\sqrt{x+h}+\sqrt{x}$ to rationalize the numerator.

Conditions: $f(x)=\sqrt{x}$.; $x > 0$ for the derivative to be finite.; $h$ is nonzero during the algebraic manipulation.

To find the derivative of $f(x)=\sqrt{x}$, substitute into the difference quotient $\frac{\sqrt{x+h}-\sqrt{x}}{h}$. Multiply the numerator and denominator by the conjugate $\sqrt{x+h}+\sqrt{x}$ to rationalize the numerator.

Conditions: $f(x)=\sqrt{x}$.; $x > 0$ for the derivative to be finite.; $h$ is nonzero during the algebraic manipulation.