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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