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Power rule
The derivative of a power function is
and we're going to calculate this here with mathematical induction.
Explanation
To determine the formula, we write the derivative of the first power as
and the derivative of the second power as
We guess that in general applies
Now comes the step from n to n + 1 and we write therefore
Differentiation shows
The product rule gives
After substitution you find
This result also arises if you replace the n by n + 1 in the original formula for the derivative of a higher power. With this the formula
has been proven by mathematical induction.