Question

Define The Principal Branch Of Αz (For Α∈C ) By Αz:=Ezlog(Α). Use The Chain Rule To Compute The Derivative Of Αz. Where Is This derivative valid

Answer 1

The derivative of λz is computed using the chain rule and is λz log(λ). This derivative is valid except for λ = 0 or when λ is a negative real number.

Step-by-step explanation:

The principal branch of λz for λ in the complex plane (λ ∈ C) can be defined by the formula λz := ez log(λ). When taking the derivative of λz, we can apply the chain rule for differentiation.

Let f(z) = λz and suppose we want to differentiate this with respect to z. Following the chain rule:

• First, differentiate eu where u = z log(λ) to get eu du/dz.
• Then, differentiate u = z log(λ) with respect to z to get log(λ) as log(λ) is a constant with respect to z.
• Combining the results gives us the derivative dλz/dz = ez log(λ) log(λ), which simplifies back to λz log(λ).

This derivative is valid except when log(λ) is undefined, which occurs when λ = 0, or when λ is a negative real number, since the logarithm of a negative number is not defined in the principal branch.