Derive Difference Quotient

[nycmath]

See attachment showing how the rule is derived.

[Math Tutor]

Nice figure - that picture is really the whole derivation.

Take two points on the curve: [tex](x,\,f(x))[/tex] and [tex](x+h,\,f(x+h))[/tex]. The slope of the secant line through them is rise over run:

[tex]\frac{\Delta y}{\Delta x}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h}[/tex]

The run collapses to just [tex]h[/tex], which is why the denominator is so simple. That quotient is the average rate of change of [tex]f[/tex] on [tex][x,\,x+h][/tex].

Now let the second point slide toward the first, i.e. [tex]h\to 0[/tex]. The secant tilts into the tangent, and the difference quotient becomes the derivative:

[tex]f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}[/tex]

Quick check with [tex]f(x)=x^{2}[/tex]:

[tex]\frac{(x+h)^{2}-x^{2}}{h}=\frac{2xh+h^{2}}{h}=2x+h\ \xrightarrow[h\to 0]{}\ 2x[/tex]

[nycmath]

Nice figure - that picture is really the whole derivation.

Take two points on the curve: [tex](x,\,f(x))[/tex] and [tex](x+h,\,f(x+h))[/tex]. The slope of the secant line through them is rise over run:

[tex]\frac{\Delta y}{\Delta x}=\frac{f(x+h)-f(x)}{(x+h)-x}=\frac{f(x+h)-f(x)}{h}[/tex]

The run collapses to just [tex]h[/tex], which is why the denominator is so simple. That quotient is the average rate of change of [tex]f[/tex] on [tex][x,\,x+h][/tex].

Now let the second point slide toward the first, i.e. [tex]h\to 0[/tex]. The secant tilts into the tangent, and the difference quotient becomes the derivative:

[tex]f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}[/tex]

Quick check with [tex]f(x)=x^{2}[/tex]:

[tex]\frac{(x+h)^{2}-x^{2}}{h}=\frac{2xh+h^{2}}{h}=2x+h\ \xrightarrow[h\to 0]{}\ 2x[/tex]

Your reply is awesome. You and Eigenvalue are right on target.