The Power Rule of Derivatives
What is a quicker way to take derivatives? 
In the prior lesson, we looked at the formal method for taking a derivative by definition, and it was quite time consuming!
On this page we are going to examine a much shorter way to take the derivative in Math.
Power Rule Formula
For any real number, n : $ \frac{d}{dx} x^n = $ 
Examples of The Power Rule
$ \frac{d}{dx} x^5 = $ 
How do we deal with coefficients?

$ \frac{d}{dx} 3x^8 = $ 

$ \frac{d}{dx}  \sqrt{2}x^7 = $ 
How do we deal with Rational Exponents?

$ \frac{d}{dx} \sqrt[3]{x^2} = $ 
How about negative exponents?

$ \frac{d}{dx} x^{4} = $ 
Lastly, what about irrational numbers?

$ \frac{d}{dx} x^{\sqrt{2}} = $ 
Practice Problems
Apply the power rule to find the derivative of the expressions below.
Problem 1) $ \frac{d}{dx} x^7 $ 
$ = 7x^{71} = 7x^6$ 
Problem 2) $ \frac{d}{dx} 3x^7 $ 
$ = 3\times7x^{71} = 21x^6$ 
Problem 3) $ \frac{d}{dx} \sqrt{2}x^{12} $ 
$ = 12\sqrt{2}x^{121}= 12\sqrt{2}x^{11} $ 
Problem 4) $ \frac{d}{dx} \sqrt{5}x^{6} $ 
$ = 6\sqrt{5}x^{61}= 6\sqrt{5}x^{7} $ 
Problem 5) $ \frac{d}{dx} \frac{2}{3x^8} $ 
$ = \frac{d}{dx} \frac{2}{3} x^{8} \\ =\frac{2}{3} \times 8 x^{81} = \frac{16}{3}x^{9} $ This is like the example above. Just remember your rules for working with negative exponents. 
Problem 6) $ \frac{d}{dx} \sqrt[5]{x^2} $ 
$ = \frac{d}{dx}x^{\frac{2}{5}} =\frac{5}{2}x^{\frac{2}{5}1} =\frac{2}{5}x^{\frac{3}{5}} = \frac{2}{3 \sqrt[5]{x^3}} $ Remember your rules for working with rational (fraction) exponents and negative exponents. 
Problem 7) $ \frac{d}{dx} x^{\pi} $ 
$ = \pi x^{\pi1} $ This is like the last example above . Remember irrational numbers and integers are not like terms so..you can't do much with this one. 
The Product Rule