proof of the power rule
Proof for all positive integers
The power rule has been shown to hold for and . If the power rule is known to hold for some , then we have
Thus the power rule holds for all positive integers .
Proof for all positive rationals
Let . We need to show
The proof of this comes from implicit differentiation.
By definition, we have . We now take the derivative with respect to on both sides of the equality.
Proof for all positive irrationals
Proof for negative powers
We again employ implicit differentiation. Let , and differentiate with respect to for some non-negative . We must show
By definition we have . We begin by taking the derivative with respect to on both sides of the equality. By application of the product rule we get
|Title||proof of the power rule|
|Date of creation||2013-03-22 12:28:06|
|Last modified on||2013-03-22 12:28:06|
|Last modified by||mathcam (2727)|