Question 1131248
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ =\ x^3\ -\ 2x^2\ +\ 7x\ -\ 3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ +\ \Delta{y}\ =\ \(x\ +\ \Delta{x}\)^3\ -\ 2\(x\ +\ \Delta{x}\)^2\ +\ 7\(x\ +\ \Delta{x}\)\ -\ 3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ +\ \Delta{y}\ =\ \(x^3\ +\ 3x^2\Delta{x}\ +\ 3x\Delta{x}^2\ +\ \Delta{x}^3\)\ -\ 2\(x^2\ +\ 2x\Delta{x}\ +\ \Delta{x}^2\)\ +\ 7\(x\ +\ \Delta{x}\)\ -\ 3]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  y\ +\ \Delta{y}\ =\ x^3\ +\ 3x^2\Delta{x}\ +\ 3x\Delta{x}^2\ +\ \Delta{x}^3\ -\ 2x^2\ -\ 4x\Delta{x}\ -\ 2\Delta{x}^2\ +\ 7x\ +\ 7\Delta{x}\ -\ 3]


Subtract original function


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \Delta{y}\ =\ 3x^2\Delta{x}\ +\ 3x\Delta{x}^2\ +\ \Delta{x}^3\ -\ 4x\Delta{x}\ -\ 2\Delta{x}^2\ +\ 7\Delta{x}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{\Delta{y}}{\Delta{x}}\ =\ \frac{3x^2\Delta{x}\ +\ 3x\Delta{x}^2\ +\ \Delta{x}^3\ -\ 4x\Delta{x}\ -\ 2\Delta{x}^2\ +\ 7\Delta{x}}{\Delta{x}}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{\Delta{y}}{\Delta{x}}\ =\ 3x^2\ +\ 3x\Delta{x}\ +\ \Delta{x}^2\ -\ 4x\ -\ 2\Delta{x}\ +\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{dy}{dx}\ =\ \lim_{\Delta{x}\right{0}}\,\frac{\Delta{y}}{\Delta{x}}\ =\ \lim_{\Delta{x}\right{0}}\,3x^2\ +\ 3x\Delta{x}\ +\ \Delta{x}^2\ -\ 4x\ -\ 2\Delta{x}\ +\ 7]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \  \frac{dy}{dx}\ =\ 3x^2\ -\ 4x\ +\ 7]
							
								
John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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