The Power Rule.

Power Rule is one of the Techniques of Differentiation.

The Power Rule states that:

The Derivative of a Function raised to n’th power is the Product of n , the function raised to the power (n-1)’th and derivative of the function raised to first power.

Mathematically we can write:

$\frac{d}{dx}\left(f(x)\right)^n = n\times (f(x))^{(n-1)} \times \frac{d}{dx}\left(f(x)\right)$

We can also further synthesize this relation and prove the rule for Derivative of general Polynomial Function. which states:

$\frac{d}{dx}\left(x\right)^n = n\times x^{(n-1)}$

Proof Of Power Rule for Natural N’th Power:

The power rule is valid for any rational “n”(eg: 4/5) power of a function. But we will prove
the power rule for only natural number power “n” here.

If “f” is a function of “x” then,
By product rule:

$\frac{d}{dx}\left(f(x)\right)^2=\frac{d}{dx}\left(f(x).f(x)\right)$

$= f(x). \frac{d}{dx}\left(f(x)\right)+f(x). \frac{d}{dx}\left(f(x)\right)=2u\frac{d}{dx}\left(f(x)\right)$

And,
$\frac{d}{dx}\left(f(x)\right)^2=\frac{d}{dx}\left(f(x)^2.f(x)\right)$

$=f(x).2f(x).\frac{d}{dx}\left(f(x)\right)+f(x)^2\frac{d}{dx}\left(f(x)\right)$

$=3.f(x).\frac{d}{dx}\left(f(x)\right)$

Similarly we can find derivative of fourth power , fifth power………. and the same result comes which is:
$\frac{d}{dx}\left(f(x)\right)^n = n\times (f(x))^{(n-1)} \times \frac{d}{dx}\left(f(x)\right)$

Use of Power Rule:

Find the derivative of: $(7x^3-4)^{3/2}$

Solution:

$\frac{d}{dx}\left(7x^3-4\right)^{3/2} = \frac{3}{2}\times (7x^3-4)^{1/2}\times \frac{d}{dx}\left(7x^3-4\right)$
$=\frac{63}{2} \dfrac{1}{2} x^2. \sqrt{7x^3-4}$

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