# The Sum Rule.

Sum Rule is one of the Techniques of Differentiation.

The Sum Rule states that:

The Derivative of  sum of two Functions is the Sum or Derivatives of the two functions.

Mathematically we can write this as:

If,

$h(x)=f(x)\pm g(x)$

Or, function “h” is the sum of functions “f” and “g”

Then ,

$h^|(x)=f^|(x)\pm g^|(x)$

Or. “function h” prime or derivative of function “h” is “function f” prime or derivative of  “f” added to “g” prime or derivative of  “g”.
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Proof of Sum Rule:

Let “p” be a fixed point then by definition of derivative:

$h^|(p)=\displaystyle\lim_{x\to p}\frac{h(x)-h(p)}{x-p}$ $=\displaystyle\lim_{x\to p}\frac{[f(x)\pm g(x)]-[f(p)\pm g(p)]}{x-p}$ $=\displaystyle\lim_{x\to p}\frac{f(x)-f(p)}{x-p} \pm \displaystyle\lim_{x\to p}\frac{g(x)-g(p)}{x-p}$ $h^|(p)=f^|(p)\pm g^|(p)$

or,

$\frac{d}{dx}\left(f(x)\pm g(x)\right) =\frac{d}{dx}\left(f(x)\right)\pm \frac{d}{dx}\left(g(x)\right)$
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Use of Sum Rule:

Find The Derivative of:  $9x^3-7x^2+5$

Solution:

$\frac{d}{dx}\left(9x^3-7x^2+5\right)=$

Using Sum Rule,

$\frac{d}{dx}\left(9x^3\right)-\frac{d}{dx}\left(7x^2\right)+\frac{d}{dx}\left(5\right)$

Using derivative of simple algebraic functions Rule,

$27x^2-14x$

so,  $\frac{d}{dx}\left(9x^3-7x^2+5\right)=27x^2-14x$

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