Use differentials to approximate the change in profit given $x=10$ units and $dx=0.2$ units, where profit is given by $P(x)=4x^{2}+90x128$.
Foundations:

A differential is a method of linearly approximating the change of a function. We use the derivative of the function at an initial point $x_{0}$ as the slope of a line, and use the standard relation

 $m\,=\,{\frac {\Delta y}{\Delta x}},$

where $\Delta y$ represents the change in $y$ values, and $\Delta x$ represents the change in $x$ values. Due to the use of the derivative $f'\left(x_{0}\right)$ as the slope, we usually rewrite this using $dy$ and $dx$ to indicate the relative changes. Thus,

 $f'(x_{0})\,=\,m\,=\,{\frac {dy}{dx}}.$

We can then rearrange this to find $dy=f'(x_{0})\cdot dx.$

Solution:
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