Math Problem Statement
prove that f(z) =(z-1)/(z+1) is differentiable at every point znot equal to -1 and find f'(z)
Solution
To prove that is differentiable at every point , and to find , we proceed as follows:
Step 1: Recall the definition of differentiability
A function is differentiable at a point if the limit
exists.
Step 2: Express as a quotient
We have . This is a quotient of two functions:
- Numerator: , differentiable everywhere.
- Denominator: , differentiable everywhere except , where .
Since the denominator is nonzero at , is well-defined and differentiable at such points, provided the quotient rule applies.
Step 3: Use the quotient rule to find
The derivative of a quotient is given by:
Here:
- , so ,
- , so .
Substitute these into the formula:
Simplify the numerator:
Step 4: Conclude differentiability
The derivative exists for all , confirming that is differentiable at every point .
Final Answer:
- is differentiable at every point .
- The derivative is:
Would you like a detailed example showing the computation of at a specific ? Let me know!
5 Related Questions:
- What does it mean for a function to be differentiable in the complex plane?
- How does the quotient rule apply in complex differentiation?
- Can be differentiable at ? Why or why not?
- How does the derivative behave as ?
- What are some real-world applications of differentiable complex functions?
Tip:
In complex analysis, always check if the denominator of a quotient is zero at a given point before concluding differentiability.
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Math Problem Analysis
Mathematical Concepts
Complex Analysis
Differentiation of Complex Functions
Quotient Rule
Formulas
f'(z) = lim(Δz → 0) [(f(z+Δz) - f(z)) / Δz]
Quotient Rule: f'(z) = [g'(z)h(z) - g(z)h'(z)] / (h(z))^2
Theorems
Definition of Differentiability in Complex Analysis
Quotient Rule for Differentiation
Suitable Grade Level
College-level or Advanced High School (Grades 11-12)
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