Calculus

From Syncleus Wiki

A great deal of electronics will require at least a year's worth of calculus to understand. This page is no substitute for a university calculus course or a calculus book (which will spend weeks and chapters on what this page handwaves through in paragraphs), but it may suffice as a reference or in a pinch.

Derivatives

The derivative of a function is the rate at which one variable is changing with respect to another. If the function is $f (x)$ , the derivative can be notated by $f'(x)$ , $\frac{d}{dx}(f(x))$ , $d f / d x$ , and if f is a function of multiple variables $f x$ or $\frac{\partial f}{\partial x}$ .

Some key properties are:

Linear: $\frac{d}{dx}(a f(x) + b g(x)) = a f'(x) + b g'(x)$

The rules for differentiating are:

Product rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$

Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$

Quotient rule: $\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$

Polynomial derivatives: $\frac{d}{dx}x^n = nx^{n-1}$

Exponential derivatives: $\frac{d}{dx}n^{u(x)} = n^u \ln(n)\frac{du}{dx}$

Logarithmic derivatives: $\frac{d}{dx}ln(u) = u' / u$

It will take a lot of thought, but this covers differentiating any functions that will be encountered in electronics. The derivatives of trigonometric functions are the derivatives of complex exponentials:

$sin(x) = \frac{e^{jx} - e^{-jx}}{-2j}$

$cos(x) = \frac{e^{jx} + e^{-jx}}{2}$

Where j (lower case) is defined as $\sqrt{-1}$

Integration

Integrating is the opposite of differentiation. Unlike with differentiation, there is no set of rules that can be followed to integrate an arbitrary function. There are many ways to think of and define integrals and integration, but some of the most useful include "area under a curve" and "accumulated change."

There are two kinds of integration, definite and indefinite. Definite integrals occur over a predefined area/region, while indefinite integrals do not. As a result, indefinite integrals have a constant of integration (usually called just c) which represents the constant on the "original function" that would have been lost when the derivative was taken.

The rules for integration are the opposite of those for differentiation, e.g.

$\int nx^{n-1} dx = x^{n+1} / (n+1) + c \rightarrow \int x^n = x^{n+1}/(n+1) + c$