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lolgarithms
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Is the power rule in calculus true for any complex exponents? i.e. does d/dz z^c = cz^(c-1) true for any c, even when c is something like i or 3i-2?
The power rule for complex exponents states that for any complex number z and any real number n, (z^n)' = n*z^(n-1).
Yes, the power rule is valid for all complex numbers, as long as the exponent is a real number.
The power rule for complex exponents is derived using the same principles as the power rule for real exponents, by applying the chain rule and the definition of complex numbers.
No, the power rule is only applicable for complex exponents with real numbers. For complex exponents with imaginary parts, the logarithmic differentiation method can be used to find the derivative.
Yes, the power rule may not apply for certain cases where the exponent is a negative integer. In such cases, the derivative must be found using the quotient rule.