Differential Calculus

Consider a function $y = f(x)$, where $x$ is the independent variable and $y$ is the dependent variable — for example, $x$ could be the position of a particle and $y=f(x)$ its velocity at that position.

Defining the derivative. Plotting $y$ against $x$ gives a curve. Let A and B be two points on the curve at $x=x_0$ and $x=x_0+\Delta x$, where $\Delta x$ is a small increment. The slope of the chord AB is $\dfrac{\Delta y}{\Delta x}$. As $\Delta x$ is made smaller and smaller (the point B moving closer to A), in the limit $\Delta x\to0$, the chord AB becomes the TANGENT to the curve at A. Although both $\Delta x$ and $\Delta y$ individually tend to zero in this limit, their RATIO need not — it tends to the slope of the tangent at $x=x_0$, called the derivative of $y$ with respect to $x$ at that point, written $\dfrac{dy}{dx}$ at $x=x_0$:

The process of finding a derivative is called differentiation.

Properties of differentiation. Let $f_1(x)$ and $f_2(x)$ be two functions of $x$ and let $s$ be a constant:

1. $\dfrac{d}{dx}\big(sf(x)\big) = s\dfrac{df(x)}{dx}$ --- (2.26)
2. $\dfrac{d}{dx}\big(f_1(x)+f_2(x)\big) = \dfrac{df_1(x)}{dx}+\dfrac{df_2(x)}{dx}$ --- (2.27) (sum rule)
3. $\dfrac{d}{dx}\big(f_1(x)\,f_2(x)\big) = f_1(x)\dfrac{df_2(x)}{dx}+f_2(x)\dfrac{df_1(x)}{dx}$ --- (2.28) (product rule)
4. $\dfrac{d}{dx}\left(\dfrac{f_1(x)}{f_2(x)}\right) = \dfrac{f_2(x)\frac{df_1(x)}{dx}-f_1(x)\frac{df_2(x)}{dx}}{f_2(x)^2}$ --- (2.29) (quotient rule)
5. If $x$ itself depends on another variable $t$ (chain rule): $\dfrac{df(x)}{dt} = \dfrac{df(x)}{dx}\cdot\dfrac{dx}{dt}$ --- (2.30)

Standard derivatives.

$\dfrac{d}{dx}(\sin x)=\cos x \quad \text{--- (2.34)} \qquad \dfrac{d}{dx}(\cos x)=-\sin x \quad \text{--- (2.35)} \qquad \dfrac{d}{dx}(\tan x)=\sec^2x \quad \text{--- (2.36)}$ …