The derivatives of a function \('f'\) at a number \('a'\) is defined as

\(\lim\limits_{h \to 0}\;\left(\dfrac{f(a+h)-f(a)}{h}\right)\) ...(1)

and is denoted by \(f'(a)\) (we are assuming that limit exists).

• This limit is used in tangent & velocity problem, therefore we give it a special name called derivatives.
•  \(f'(a)=\lim\limits_{x \to a}\;\left(\dfrac{f(x)-f(a)}{x-a}\right)\)...(2)   \(\to\) define \(x-a=h\) in (1)
• (1) and (2) are different ways of writing the derivatives.
•  Derivatives at  \(x=a=\) \(f'(a)\) = Slope of tangent to the graph of  \('f'\) at \((a,\,f(a))\)

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

• Given any limit in the form of L.H.S, identify the value of \('a'\) and the function \('f'\).
• e.g. \(=\lim\limits_{h\to 0}\,\left(\dfrac{(2+h)^3-8}{h}\right)\)
• \(=\dfrac{f(a+h)-f(a)}{h}\)

Then,  \(f(a+h)=(2+h)^3\)

\(\Rightarrow\,a=2,\;f(x)=x^3\)

\(\therefore\) Limit =\(f'(2)=3x^2\) at \(x=2\)

\(=12\)

Let \(y\) be a quantity which is a function of \(x\), \(y=f(x)\). If \(x\) changes from \(x_1\) to \(x_2\) we say.

\(\Delta x =x_2-x_1\) = change in \(x\)

Corresponding change in \(y=f(x_2)-f(x_1)=\Delta y\)

• The average rate of change of  \(y\) with respect to \(x\) is

\(=\dfrac{f(x_2)-f(x_1)}{x_2-x_1}=\dfrac{\Delta y}{\Delta x}\)

• Instantaneous rate of change of \(y\) with respect to \(x\) at \(x=x_1\) is

\(=\lim\limits_{x_2\to x_1}\;\left(\dfrac{f(x_2)-f(x_1)}{x_2-x_1}\right)\)

\(=\lim\limits_{\Delta x\to 0}\;\left(\dfrac{f(x_1+\Delta x)-f(x_1)}{\Delta x}\right)\)

Correlating the above limit with derivative we say, \(f'(a)\) is the instantaneous rate of change of \(y=f(x)\) with respect to  \(x\) when, \(x=a\).

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

e.g.  \(\lim\limits_{x\to 2}\;\dfrac{x^{10}-2^{10}}{x-2}=10×2^9\)

because \(f(x)=x^{10},\;a=2\)

\(\therefore\) \(f'(x)\) \(=10x^9\) and \(f'(2)\) \(=10×2^9\)

The equation of tangent line to  \(y=f(x)\)  at any point \(P(a,\,f(a))\) on it is a line passing through \(P(a,\,f(a))\) and whose slope is \(f'(a)\).

Consider the graph \(y=f(x)\) as shown in figure.

Different tangents at points \(P,\,Q,\;R\) are drawn as shown in figure

We see

At point \(P\)

i.e. at \(x=x_1\), tangent makes acute angle with \(x-\) axis which implies

Slope of tangent \(\to\;f'(x_1)>0\)

at \(x=x_1\)

At point \(Q\)

i.e. at \(x=x_2\), tangent is parallel to \(x-\) axis which implies

Slope of tangent at \(x=x_2\,\to\) \(f'(x_2)=0\)

At point \(R\)

i.e. at \(x=x_3\), tangent makes obtuse angle with \(x-\) axis which implies

Slope of tangent at \(x=x_3\,\to\) \(f'(x_3)<0\)

It also concludes

\(f'(x_1)>f'(x_2)>f'(x_3)\)

Let \(y=f(x)\) be a function , if \(x \) changes from \(x _1 \) to \(x _2\), the change in \(x \) (called the increment of \(x \)) is  \(\Delta x=x_2 -x_1\)

The corresponding change in \(y\) is \(\Delta y = f(x_2) - f(x_1)\)

The difference quotient = \(\dfrac{\Delta y}{\Delta x} = \dfrac{f(x_2)-f(x_1)}{x_2-x_1}\)  is called the average rate of change of \(y\) with respect to \(x\) over the interval \((x_1,x_2)\) 

• It can be interpreted as slope of secant line PQ as shown.

Average velocity is the average rate of change of position with respect  to time during an interval \((t_1,t_2)\) .

Average Velocity = \(\dfrac{\Delta x}{\Delta t} = \dfrac{x(t_2)-x(t_1)}{t_2-t_1}\) 

where x=position,t=time