# THE POWER RULE OF DIFFERENTIATION

The power rule of differention states that:

$\frac{d}{dx}(ax^n)=n(ax^{n-1})$

If we are to express the above formula in words, the power rule states that you should multiply the coefficient of x by the exponent n and then subtract one from the exponent.

Example 1

Find the derivative of $16x^2$

Solution:

The power of x is 2,  therefore

$$2(16x^{2-1})=32x$$

Example 2

Find the derivative of $32x^3+16x^2+4x$

Solution:

$$3(32x^{3-1})+ 2(16x^{2-1})+4(x^{1-1})$$

$$96x^2+32x+4$$

Example 3

Find the derivative of

$\frac{3}{x^4}+\frac{27}{x^2}-\frac{3}{x^3}$

Solution:

This can be rewritten as

$$3x^{-4}+27x^{-2}-3x^{-3}$$

$$-4(3x^{-4-1})-2(27x^{-2-1})+3(x^{-3-1})$$

$$12x^{-5}-54x^{-3}+9x^{-4}$$

if you like, you can go further

$$\frac{-12}{x^5}-\frac{54}{x^5}+\frac{9}{x^4}$$

Example 4

Find the derivative of $27x^2+\frac{1}{x^4}+\frac{27}{x^3}$

Solution:

The above expression can be rewritten as

$$27x^2+x^{-4}+27x^{-3}$$

Using the power rule

$$2(27x^{2-1})-4(x^{-4-1})-3(27x^{-4})$$

$$54x-4x^{-5}-81x^{-4}$$

If you like

$$54x-\frac{4}{x^5}-\frac{81}{x^4}$$

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