\(f'{(a)}\) \(=\lim\limits_{h\to 0}\;\left(\dfrac{f(a+h)-f(a)}{h}\right)\)

Now let \('a'\) be a variable number \(x\), so that as \(x\) changes this expression changes with \(x\) and hence becomes a function of \(x\).

\(f'{(x)}\) \(=\lim\limits_{h\to 0}\;\left(\dfrac{f(x+h)-f(x)}{h}\right)\)

We say that this is derivative of f( i.e., f'), provided the limit exists at that \(x\).

• It is the slope of tangent to the graph of \(f(x)\) at the point \((x,\,f(x))\).
• The domain of \(f'\) may be smaller than the domain of \(f\).

(There may be some values of \(x\) for which \(f'\) may not be defined or limit does not exist).

We normally take \(y\) as the dependent variable and \(x\) as the independent variable.

\(y=f(x)\)

We use the notation:

derivatives of \(y\) with respect to \(x=f'(x)\)

Sometimes we use the notations,

\(f'(x)=y'=\underbrace{\dfrac{dy}{dx}=\dfrac{df}{dx}=\dfrac{df(x)}{dx}=Df(x)=D_xf(x)}_\text{All expression have the same meaning}\)

• \(D\) and \(\dfrac{d}{dx}\) are called differentiation operators, because they indicate operation of differentiation.
• The process of finding derivatives is called differentiation.
• \(\dfrac{dy}{dx}\Bigg|_{x=a}\) = \(f'{(a)}\)is the other notation used.

Note that \(f'{(a)}\) means first take the derivative of the function with respect to \(x\)and then put \(x=a\) in the derivative expression. It is not the derivative of \(f(a)\), which will always be 0 because \(f(a)\) is a constant.

• Continuity is a necessary but not sufficient condition for differentiable at a point \(x=a\).

i.e. if a function is continuous at \(x=a\), then it may or may not be differentiable at \(x=a\).

• If a function is differentiable at \(x=a\), then it must be continuous at \(x=a\).
• A broken graph will not allow a tangent to be drawn and hence no slope, so no \(f'\).

• A function \('f'\) is differentiable in an open interval \((a,\,b)\) or \((a,\,\infty)\) or \((-\infty,\,a )\) if it is differentiable at each and every point in the interval.
• If there is a non differentiable point in the interval \((a,\,b)\) then the function is said to be non differentiable in the entire interval.

\(f(x)=|x|\) is non differentiable at \(x=0\), in general \(|g(x)|\) will be non differentiable at all the roots of \(g(x)=0\).

A function \('f'\) is said to be differentiable at \(x=a\) if \(f'{(a)}\) exists i.e the limit of definition of derivative exists.

i.e.

\(=\lim\limits_{h\to 0}\;\left(\dfrac{f(a+h)-f(a)}{h}\right)\) exists.

• Among commonly used functions, \(f(x)=|x-a|\) is non differentiable at \(x=a\) because R.H.L. \(\neq\) L.H.L. for derivative limit.
• In general \(g(x)=|f(x)|\) is non differentiable at all those values of \(x\) where \(f(x)=0\).