derivatives as linear approximations

The standard definition of the derivative at \[c\] is \[f^{\prime}(c)=\lim_{h\to0}\frac{f(c+h)-f(c)}{h}\], which can also be written as \[\lim_{h\to0}\frac{f(c+h)-f(c)-f^{\prime}(c)h}{h}=0\]. Now replace the limit with \[\frac{f(c+h)-f(c)-f^{\prime}(c)h}{h}=\epsilon(h)\] such that as \[h\to0\], \[\epsilon(h)\to0\]. Now, let \[r(h)=\epsilon(h)\cdot h\], then \[f(c+h)-f(c)-f^{\prime}(c)h=r(h)\]. Then, our original limit can be translated into \[\lim_{h\to0}\frac{r(h)}{h}=0\].

Therefore, another way to view the definition of derivative is to write \[f(c+h)=f(c)+f^{\prime}(c)h+r(h)\], where \[f(c+h)\] is linearly approximated with \[f(c)+f^{\prime}(c)h\] and the remainder \[r(h)\] (or sometimes written as \[o(h)\] when \[h\to0\]). This means that the \[f\] value of a point near \[x=c\] can be approximated using the tangent line at \[c\].

We can also show this geometrically. The graph of \[f\] around point \[c\] is almost the same as the tangent at point \[c\] if we zoom in far enough. To use this to our advantage, when a point \[x\] is near \[c\], we can write it as \[f(x)\approx f(c)+f^{\prime}(c)(x-c)\]. Assuming the coordinates \[\left( x_{0},y_{0} \right)=\left( c,f(c) \right)\] and the tangent line is equal to \[f\], since \[m=\frac{y-y_{0}}{x-x_{0}}\], then \[f^{\prime}(c)=\frac{f(x)-f(c)}{x-c}\implies f(x)=f^{\prime}(c)(x-c)+f(c)\].

Now that we've established the origin of the approximation \[f(x)\approx f(c)+f^{\prime}(c)(x-c)\], if we replace \[x\] with \[c+h\] (meaning any point near \[c\] such that it's distance is \[h\] away from \[c\]), then \[f(c+h)\approx f(c)+f^{\prime}(c)(c+h-c)\implies f(c+h)-f(c)\approx f^{\prime}(c)h\]. If we rearrange it, we get the original equation, \[f(c+h)-f(c)=f^{\prime}(c)h+r(h)\], as \[h\to0\].

\[f^{\prime}(c)\] in the equation above may also be intepreted as a scaling factor of points in \[f\] that are near \[x=c\], as when we move \[h\] distance away from \[c\], then \[f(c+h)\] would be \[h\cdot f^{\prime}(c)\] away from \[f(c)\].