Calculus with polynomials
In mathematics, polynomials are perhaps the simplest functions with which to do calculus. Their derivatives and integrals are given by the following rules:
Hence the derivative of x100 is 100x99 and the integral of x100 is x101/101 + c.
Proof
Because differentiation is linear, we have:
So it remains to find  for any natural number r. The derivative of function f(x) is given by Newton's difference quotient
By the binomial theorem, and using the C-notation of combinations,
and therefore
The derivative is the limit of this as 
which gives the claimed result.
Generalisation
is generally true for all values of k where xk is meaningful. In particular it holds for all rational k for values of x where xk is defined.
Similarly for integration, see table of integrals.
References
- Calculus of a Single Variable: Early Transcendental Functions (3rd Edition) by Edwards, Hostetler, and Larson (2003) ISBN 0618226877
Referenced By
Calculus | Chain rule | Derivative | Derivative (calculus) | Differentation | Differentiable function | Differential calculus | List of calculus topics | List of mathematical proofs | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics | List of polynomial topics | List of proofs | Taylor's theorem | Taylors theorem
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