Power rule, Product rule, Quotient rule, Chain rule
Lists of integrals
parts, disks, cylindrical
partial fractions, changing order
In mathematics, polynomials are perhaps the simplest functions used in calculus. Their derivatives and indefinite integrals are given by the following rules:
Hence, the derivative of x100 is 100x99 and the indefinite integral of x100 is where C is an arbitrary constant of integration.
This article will state and prove the power rule for differentiation, and then use it to prove these two formulas.
The power rule for differentiation states that for every natural number n, the derivative of is that is,
The power rule for integration
for natural n is then an easy consequence. One just needs to take the derivative of this equality and use the power rule and linearity of differentiation on the right-hand side.
To prove the power rule for differentiation, we use the definition of the derivative as a limit:
Substituting f(x) = xn gives
Since the numerator is a difference of powers, it factors:
We can simplify the first factor:
Then we can cancel the h:
Once the h has been canceled, we may evaluate the limit by substituting h=0.
Each summand is now the same xn − 1 so:
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