Derivatives

The first derivative \(f'(x)\) of a function \(f(x)\) is defined as the limit \[ \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\] At a particular value \(a\), we can also define \(f'(a)\) as \[ f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\] if this limit exists. This limit will be equal to the slope of the tangent line to \(f(x)\) at \(x=a\).

The second derivative \(f''(x)\) is found by taking the derivative of the first derivative. That is, the value of \(f''(x)\) is the slope of the tangent line to the graph of \(f'(x)\) at \(x\).