**Approximation by Differentials**

A method for approximating the value of a function near a known value. The method uses the tangent line at the known value of the function to approximate the function's graph. In this method Δ*x* and Δ*y* represent the changes in *x* and *y* for the function, and *dx* and *dy* represent the changes in *x* and *y* for the tangent line.

Example: | Approximate \(\sqrt {10} \) by differentials. |

Solution: | \(\sqrt {10} \) is near \(\eqalign{\sqrt {10} &= f\left( {x + \Delta x} \right)\\ &\approx f\left( x \right) + f'\left( x \right)\Delta x\\ &= \sqrt x + \frac{1}{{2\sqrt x }}\Delta x\\ &= \sqrt 9 + \frac{1}{{2\sqrt 9 }}\left( 1 \right)\\ &= 3\frac{1}{6}}\). Thus we see that \(\sqrt {10} \approx 3\frac{1}{6} = 3.166\bar 6\). This is very close to the correct value of \(\sqrt {10} \approx 3.1623\). |

**See also**

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