Differentiation and Derivation Techniques

Instantaneous Rates

Average vs. Instantaneous

Calculus is the study of change and rates are the fundamental unit to measure. But students have been familiar with rates at the start of their abstract math studies; the \(m\) in \(y = mx + b\) is up there with the mitochondria as a memorable high school trope. Calculating the slope is one of those wonderful cases where the textual definition intuitively matches its mathematical setup: the change in the output divided by the change in the input, \[m = {\Delta y \over \Delta x}\] Or if using function notation with a change of \(h\): \[m = {f(x + h) - f(x) \over h}\]

That setup works well for linear situations, but it doesn’t capture the full story for anything changing at a dynamic rate. For example, consider an object falling from 100 meters up! Its ideal trajectory would be modeled as \(y(t) = -9.81t^2 + 100\). Consider what happens in the first second, from \(t = 0 s\) to \(t = 1 s\). It will go from \(100\; m\) to \( 90.19\; m\) for a change of \(-9.81 {m \over s^2}\). But then what if we redo the math from \(t = 0 s\) to \(t = 2 s\)? It results in a change of \(-19.62 {m \over s^2}\). We can still learn the average value for the rate of change (which in this case is velocity), but it would fail to show how the object is accelerating towards the ground and that the rate is changing from one instant to the next.

There are a lot of situations where the average rate is adequate, especially in the case of real world data collection where those moments in between would only ever be inferred. But if an instantaneous rate can be found, it can be a remarkably powerful tool for analyzing how something behaves and changes over time. An instant would be a situation where the rate of change is calculated over an infinitesimal amount, as the value of \(h\) is essentially \(0\). Since directly using \(0\) would lead to an indeterminate form, we instead calculate instantaneous change using limits: \[\lim_{h\to 0} {f(x + h) - f(x) \over h}\] The above limit still has the slope in its DNA and serves a similar role for non-linear relations. This calculation will also represent the slope of a tangent line of the original function at whatever \(x\) value is being used!

The Derivative

Derivatives and Continuity

Infinitesimals and the Bishop Berkeley

Derivation Techniques

Derivation Rules

Transcendental Derivatives

Products and Quotient Derivatives

Higher Order Derivatives

Implicit Differentiation and Inverse Functions

The Chain Rule

Derivatives of Inverse Functions

Other Resources

TBD