Rule No. 1: If y = xn, then = nxn-1That’s it. Wasn’t that simple? Of course, this and all of the following rules can be derived easily from the
definition of the derivative. Look at these next few examples of the Power Rule in action.
Notice that when the power of the function is negative, the
power of the derivative is more negative.Example 1: If y = x^5 , then = 5x^4.
Example 2: If y = x^20 , then = 20x^19.
When the power is a fraction, you should be careful to get the
subtraction right (you’ll see the powers , , , − , and −often, so be comfortable with subtracting 1 from them).Example 3: If f(x) = x−5, then f′(x) = −5x−6.
Example 4: If u = x, then .
Example 5: If y = x^1 , then = 1x^0 = 1. (Because x^0 is 1!)
Example 6: If y = x^0 , then = 0.
When the power is 1, the derivative is just a constant. When
the power is 0, the derivative is 0.This leads to the next three rules.
Rule No. 2: If y = x, then = 1