Let us see some of the rules to find the derivatives of a function.

Let f(x) and g (x) be the two functions where x $\epsilon$ R.

### Sum Rule:

The derivative of sum of the functions, is the sum of derivative of the derivatives of each function.

(i. e),

$\frac{d}{dx}$ [ f (x) + g (x) ] =

$\frac{d}{dx}$ [ f (x) ] +

$\frac{d}{dx}$ [ g (x) ]

In general,

$\frac{d}{dx}$ [ u + v + w + ............... ] =

$\frac{du}{dx}$ +

$\frac{dv}{dx}$ +

$\frac{dw}{dx}$ + . . . . . . . . . . . . . .

### Difference Rule:

The derivative of difference of the functions, is the differenceof derivative of the derivatives of each function.

(i. e),

$\frac{d}{dx}$ [ f (x) - g (x) ] =

$\frac{d}{dx}$ [ f (x) ] -

$\frac{d}{dx}$ [ g (x) ]

In general,

$\frac{d}{dx}$ [ u - v - w - ............... ] =

$\frac{du}{dx}$ -

$\frac{dv}{dx}$ -

$\frac{dw}{dx}$ - . . . . . . . . . . . .

### Product Rule:

The derivative of the product of the functions is found by the following method.

$\frac{d}{dx}$ [ f (x). g (x) ] = f (x) .

$\frac{d}{dx}$[ g (x) ] + g (x) .

$\frac{d}{dx}$ [ f (x) ]

The above can also be stated as

$\frac{d}{dx}$ [ u. v ] = u .

$\frac{dv}{dx}$ + v.

$\frac{du}{dx}$The above rule can be extended to the product of any finite number of functions, as follows.

$\frac{d}{dx}$ [ u. v. w ] = v. w.

$\frac{du}{dx}$ + u . w .

$\frac{dv}{dx}$ + u . v

$\frac{dw}{dx}$ ### Quotient Rule:

If u and v are functions of x and v (x) $\ne$ 0, then

$\frac{\mathrm{d} }{\mathrm{d} x}\left (\frac{u}{v} \right)$ =

$\frac{v.\frac{\mathrm{du}}{\mathrm{d} x}-u.\frac{\mathrm{dv} }{\mathrm{d} x} }{v^{2}}$### Chain Rule - Differentiation of Composite functions:

In the case of composite function, where y = (f o g) (x), then

$\frac{dy}{dx}$ = $\frac{d}{dx}$ [ (f o g) (x) ] = f ' [ g (x) ] g ' (x)

Also, if y = f (u) , u = g (v) and v = h (x), then

$\frac{dy}{du}$ = f ' (u),

$\frac{du}{dv}$ = g ' (v)

and $\frac{dv}{dx}$ = h ' (x)

Therefore, $\frac{dy}{dx}$ = $\frac{dy}{du}$ . $\frac{du}{dv}$ . $\frac{dv}{dx}$

= f ' (u) . g ' (v) . h ' (x)

= f ' [ g { h (x) } ] . g ' { h (x) } . h ' (x)