Exercise · ZW-COMPLEX-DERIVATIVE-008 · ★★☆ Intermediate
Compute \(h'(1)\) for \(h(z)=\dfrac{z^2}{z+1}\).
Choose one option.
Show answer
- D. \(\dfrac34\)
Explanation
Quotient rule: \(h'(z)=\frac{2z(z+1)-z^2}{(z+1)^2}=\frac{z^2+2z}{(z+1)^2}\), which equals \(\frac34\) at \(z=1\).
Review the concept: Complex derivative
The derivative \(f'(z_0)\) of a complex function \(f\) at a point \(z_0\) is the limit of the difference quotient as \(z\) approaches \(z_0\) from any direction in the plane, if that limit exists.
\[f'(z_0)=\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}\]
More practice problems: complex derivative
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- ZW-COMPLEX-DERIVATIVE-004 What is the derivative of e^z at z=0?
- ZW-COMPLEX-DERIVATIVE-005 If f is differentiable at z_0, what follows?
- ZW-COMPLEX-DERIVATIVE-006 Compute f'(1+i) for f(z)=z^3-2z^2+5z.
- ZW-COMPLEX-DERIVATIVE-007 Compute g'(0) for g(z)=z e^z.
- ZW-COMPLEX-DERIVATIVE-009 For f(z)=bar z at z_0=2, what happens to the difference quotient?