Power Rule in Differentiation (Part 2)
How to solve the power rule in differentiation problems (when the exponent is a non-zero integer): formula, proof, example, and its solution.
Previously, you've learned that
[xn]' = nxn - 1
when n is a natural number.
Power rule in differentiation (Part 1)
This is also true
when n is a non-zero integer: -1, -2, -3, ... .
Set n = -m.
Then [xn]' = [x-m]'.
Reciprocal rule in differentiation
Change -m to n.
Then [xn]' = nxn - 1.
Divide each term on the numerator by x3.
Then y = 6x3 - 3 + 2x-1 - x-3.
Use the power rule to find y'.
y' = 6⋅3x2 - 0 + 2⋅(-1)x-2 - (-3)x-4
Change the negative exponents into the fractions.
Then y' = 18x2 - 2/x2 + 3/x4.