Power rule, Product rule, Quotient rule, Chain rule
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Integration by:
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In mathematics, polynomials are perhaps the simplest functions used in calculus. Their derivatives and indefinite integrals are given by the following rules:
and
Hence, the derivative of x^{100} is 100x^{99} and the indefinite integral of x^{100} 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.
Contents
Power rule
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 righthand side.
Proof
To prove the power rule for differentiation, we use the definition of the derivative as a limit:
Substituting f(x) = x^{n} 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 x^{n − 1} so:
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