Calculus
Table of Contents
Differentiation Shortcuts
Sum Rule: \(\frac{d}{dx}(u + v) = \frac{du}{dx} + \frac{dv}{dx} = u'+v'\)
Product Rule: \(\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx} = uv'+vu'\)
Quotient Rule: \(\frac{d}{dx}(\frac{u}{v}) = \frac{v(\frac{du}{dx})-u(\frac{dv}{dx})}{v^{2}} = \frac{vu'-uv'}{v^{2}}\)
Chain Rule: \(\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}\)
Max & Mins: ???
Trigonometric
Logarithmic
\[\frac{d}{dx}lnx = \frac{1}{x}\]Differential Equations
A differential equation is one that relates a function to its derivative. For example, if you have a model where the rate of change of \(B\) is proportional to \(B\), then the derivative of \(B\) is proportional to \(B\) by some constant \(k\):
\[\frac{dB}{dt} = kB\]Exponential Model
The solution to a differential equation for an exponential model is
\[B(t) = C \cdot e^{kt}\]