In calculus, sometimes generalization makes a problem easier instead of harder. This is called as paradox of generalization.

For any function \(f\), the derivative of \(f\) is the function \(f'(x)\) defined by the formula

\(f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}\)

• The domain of \(f'\) is the set of all values of \(x\) for which this limit is defined.
• If \(a\) is in the domain of \(f'\), then we say that \(f\) is differentiable at \(a\).
• \(f'(a)\) is the rate of change of \(f(x)\) with respect to \(x\) at \(x=a\), which is also the slope of the line tangent to the graph of \(f\) at the point \((a,f(a))\).
• The process of finding the derivative of a function is called differentiation.
• If \(I\) is an open interval, we say that \(f\) is differentiable on \(I\) if it is differentiable at \(a\) for every number \(a \in I\).
• The formula that defines \(f'(x)\) contains both \(f(x)\) and \(f(x+h)\), in order for \(f'(x)\) to be defined, not only must \(f(x)\) be defined, but also \(f(x+h)\) mus be defined for \(h\) close to \(0\). In other words, there must be some number \(d>0\) such that the interval \((x-d, x+d)\) is contained in the domain of \(f\).

Suppose \(f\) is differentiable at \(a\). Then

\(f'(a) = \lim_{x \to a} \dfrac{f(x)-f(a)}{x-a}\)

If \(f\) is differentiable at \(a\), then \(f\) is continuous at \(a\).

\(\dfrac{dy}{dx} = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x} = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}=f'(x)\)

• The notation \(dy/dx\) was introduced by Leibniz, so it is sometimes called Leibniz notation for the derivative.
• The \(f'\) notation was introduced by Joseph Louis Lagrange.

\(\dfrac{dy}{dx}\Bigr|_{x=a}\)

• The above notation is used to denote the derivative of \(y\) with respect to \(x\), with the number \(a\) substituted for \(x\).

• Intuitive definition