# Compute \Delta y and dy for the given values of x and dx=\Delta x. (Round your answer to three...

## Question:

Compute {eq}\Delta y{/eq} and {eq}dy{/eq} for the given values of {eq}x{/eq} and {eq}dx=\Delta x{/eq}. (Round your answer to three decimal places.)

{eq}y=\sqrt x, \ x=1, \ \Delta x=1{/eq}

## Derivatives:

The derivatives of functions can be used for several applications. We can use differentials to basically acquire the rates of change of functions. Furthermore, we can apply differentials to determine the error associated with a computed variable from a measurement with inaccuracy.

For the given function, we simply need to apply the differential and then plug in the given values to determine {eq}\displaystyle \Delta y {/eq}. We proceed with the following solution.

{eq}\begin{align} \displaystyle y &= \sqrt x\\ \frac{dy}{dx} &= \frac{d}{dx} x^ \frac{1}{2}\\ \frac{dy}{dx} &= \frac{1}{2}x^{- \frac{1}{2}}\\ \frac{dy}{dx} &= \frac{1}{2\sqrt{x}}\\ dy &= \frac{dx}{2\sqrt{x}}\\ \text{We take the total differential.}\\ \Delta y &= \frac{\Delta x}{2\sqrt x}\\ \text{We substitute the given values.}\\ \Delta y &= \frac{1}{2\sqrt 1}\\ \Delta y &= \frac{1}{2}\\ &= 0.500 \end{align} {/eq}