What is Calculus?

Calculus is the study of change, with the basic focus being on:

1. Rate of change (Differential Calculus)
2. Accumulation (Integral Calculus)

Notations for the derivative

• In functional notation, we can write a function \(y\) in terms of \(x\) as \(y=f(x)\).
• We read \(y = f(x) \) as \(y\) is a function of \(x\).
• There are a number of different notations for the derivative of a function, the two most common are: \(\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}\) or \(f’(x)\).
• The average rate of change of a function is determined from the gradient of a secant between the points, \((x, f(x))\) and \((x+h,f(x+h))\) is: 
  \[ \mbox{Average rate of change} =\frac{\Delta f}{\Delta x}= \frac{f(x+h)-f(x)}{h}\]
• Note the Greek symbol Delta, \(\Delta\) represents change in, so \(\Delta x\) reads as change in \(x\).
• The instantaneous rate of change of a function is termed the derivative, which is the limit as \(h\) (or step size) approaches \(0\):
  \[ f’(x) = \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h} \]
• The derivative is also the gradient of the tangent line.