Hi Scholars!

Let’s revisit a concept from our first calculus course: the derivative.

Definition: A functionfis differentiable at a pointzprovided the limit of (f(x)-f(z))/(x-z) exists as x approaches z. The limit is called the derivative offatz.

Admittedly, this is a concept that I only “sort of “ grasped the first time around, and in an extremely “hand wavy” fashion. As we can see in the graphic above, the key to formalizing the concept is to note how the slope of secant lines, shown in green, get “close” to the derivative (*i.e.* within ε, the red lines) whenever secant lines are drawn through points *x *sufficiently close to *z *(*i.e. *within δ) *.*

How would we go above using an ε-δ argument to prove some value is, in fact, the derivative? When I am tasked with showing the derivative of a function *f* at a point *z* is *L*, here’s a breakdown on how I might proceed.

State the definition of what must be shown,

*e.g.*“Let ε > 0 be given. It suffices to show there is δ > 0 such that if 0 < |x - z| < δ, then | (

*f*(x) -*f*(z))/(*x*-*z*) -*L*| < ε.”Establish some bound on | (

*f*(x) -*f*(z))/(*x*-*z*) -*L*| as a function of |*x*-*z*|.Pick δ such that if |

*x*-*z*| were replaced in the bound in the last step by δ, then the whole expression would be less than ε.Explicitly show this choice of δ makes that expression less than ε.

Upon completing Step 4, the definition in Step 1 has been verified. If we want to show *f* is differentiable on, say, an interval (*a*,*b*), then we must do the above steps with an arbitrary point *z* in (*a,b*), also replacing *L* with the formula we have for the derivative.

Stay Awesome.

Howard