Use differentials to find $dy$ given $y=x^{2}6x,~x=4,~dx=0.5.$
Foundations:

When we use differentials, we are approximating a value for a function by using the slope of the derivative. The idea is that given a distance $dx$ from a point $x$, we can use $f'(x)$, the slope of the tangent line, to find the rise, $dy$. Recalling that we can write

 $f'(x)\,=\,{\frac {dy}{dx}},$

the relation is

 $dy\,=\,f'(x)\cdot dx,$

where we use the given specific $x$value to evaluate $f'(x)$.

Solution:
Step 1:

By the power rule, we have

 $f'(x)\,=\,2x6.$

We need to evaluate this at the given value $x=4$, so

 $f'(4)\,=\,2(4)6\,=\,2.$

Step 2:

We use the values given and the result from step 1 to find

 $dy\,=\,f'(x)\cdot dx\,=\,2(0.5)\,=\,1.$

Final Answer:

 $dy\,=\,1.$

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