Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/54

Second step.	$\scriptscriptstyle {y+\Delta y}$	$\scriptscriptstyle {=3x^{2}+6x\cdot \Delta x+3(\Delta x)^{2}}$	$\scriptscriptstyle {+5}$
	$\scriptscriptstyle {y}$	$\scriptscriptstyle {=3x^{2}}$	$\scriptscriptstyle {+5}$
	$\scriptscriptstyle {\Delta y}$	$\scriptscriptstyle {=6x\cdot \Delta x+3(\Delta x)^{2}}$.
Third step.	$\scriptscriptstyle {\frac {\Delta y}{\Delta x}}$	$\scriptscriptstyle {=6x+3\cdot \Delta x}$.
Fourth step.	$\scriptscriptstyle {\frac {dy}{dx}}$	$\scriptscriptstyle {=6x}$.   Ans.
We may also write this
$\scriptscriptstyle {{\frac {d}{dx}}(3x^{2}+5)}$		$\scriptscriptstyle {=6x}$.

 

Illustrative Example 2. Differentiate $\scriptscriptstyle {x^{3}-2x+7}$.

Solution.	Place	$\scriptscriptstyle {y}$		$\scriptscriptstyle {=x^{3}-2x+7}$.
First step.		$\scriptscriptstyle {y+\Delta y}$		$\scriptscriptstyle {=(x+\Delta x)^{3}-2(x+\Delta x)+7}$.
				$\scriptscriptstyle {=x^{3}}$	$\scriptscriptstyle {+3x^{2}\cdot \Delta x+3x\cdot (\Delta x)^{2}+(\Delta x)^{3}}$	$\scriptscriptstyle {-2x}$	$\scriptscriptstyle {-2\cdot \Delta x}$	$\scriptscriptstyle {+7}$.
Second step.			$\scriptscriptstyle {y+\Delta y}$	$\scriptscriptstyle {=x^{3}}$	$\scriptscriptstyle {+3x^{2}\cdot \Delta x+3x\cdot (\Delta x)^{2}+(\Delta x)^{3}}$	$\scriptscriptstyle {-2x}$	$\scriptscriptstyle {-2\cdot \Delta x}$	$\scriptscriptstyle {+7}$
			$\scriptscriptstyle {y}$$\scriptscriptstyle {+\Delta y}$	$\scriptscriptstyle {=x^{3}}$		$\scriptscriptstyle {-2x}$		$\scriptscriptstyle {+7}$
			$\scriptscriptstyle {\Delta y}$	$\scriptscriptstyle {=}$	$\scriptscriptstyle {3x^{2}\cdot \Delta x+3x\cdot (\Delta x)^{2}+(\Delta x)^{3}}$		$\scriptscriptstyle {-2\cdot \Delta x}$
Third step.		$\scriptscriptstyle {\frac {\Delta y}{\Delta x}}$		$\scriptscriptstyle {=3x^{2}+3x\cdot \Delta x+(\Delta x)^{2}-2.}$
Fourth step.		$\scriptscriptstyle {\frac {dy}{dx}}$		$\scriptscriptstyle {=3x^{2}-2}$.   Ans.
Or,	$\scriptscriptstyle {{\frac {d}{dx}}(x^{3}-2x+7)}$			$\scriptscriptstyle {=3x^{2}-2}$.

 

Illustrative Example 3. Differentiate $\scriptscriptstyle {\frac {c}{x^{2}}}$.

Solution.	Place	$\scriptscriptstyle {y}$	$\scriptscriptstyle {={\frac {c}{x^{2}}}}$.
First step.		$\scriptscriptstyle {y+\Delta y}$	$\scriptscriptstyle {={\frac {c}{(x+\Delta x)^{2}}}}$.
Second step.		$\scriptscriptstyle {y+\Delta y}$	$\scriptscriptstyle {={\frac {c}{(x+\Delta x)^{2}}}}$
		$\scriptscriptstyle {y}$$\scriptscriptstyle {+\Delta y}$	$\scriptscriptstyle {={\frac {c}{x^{2}}}}$
$\scriptscriptstyle {\Delta y}$			$\scriptscriptstyle {={\frac {c}{(x+\Delta x)^{2}}}-{\frac {c}{x^{2}}}={\frac {-c\cdot \Delta x(2x+\Delta x)}{x^{2}(x+\Delta x)^{2}}}}$.
Third step.		$\scriptscriptstyle {\frac {\Delta y}{\Delta x}}$	$\scriptscriptstyle {=-c\cdot {\frac {2x+\Delta x}{x^{2}(x+\Delta x)^{2}}}}$.
Fourth step.		$\scriptscriptstyle {\frac {dy}{dx}}$	$\scriptscriptstyle {=-c\cdot {\frac {2x}{x^{2}(x)^{2}}}}$.
			$\scriptscriptstyle {=-{\frac {2c}{x^{3}}}}$.   Ans.
Or,	$\scriptscriptstyle {{\frac {d}{dx}}\left({\frac {c}{x^{2}}}\right)}$		$\scriptscriptstyle {={\frac {-2c}{x^{3}}}}$.