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
$$54x-\frac{4}{x^5}-\frac{81}{x^4}$$
There you have it. If you found this post helpful, do well to share it with your friends.