How to Compute the Derivative of a Complex Modulus Function

[sigmund]

I have to calculate the derivative of this function:

$$f(t)=\vert\hspace{0.07cm}u(t)+i\cdot{}v(t)\vert$$

The derivative should be expressed with u, u', v and v'.
How do you calculate this derivative?

 

[Nexus[Free-DC]]

Okay, this is a combination of the chain rule and implicit differentiation.

The first thing to do is let $$a=u(t)+iv(t)$$

Now let $$f(x)=\sqrt{a^2}$$ and the derivative becomes

$$\frac{df}{dt}=\frac{df}{da}\frac{da}{dt}$$

You should be able to proceed from there. If not, yell out.

 

[Gokul43201]

I have to calculate the derivative of this function:

$$f(t)=\vert\hspace{0.07cm}u(t)+i\cdot{}v(t)\vert$$

The derivative should be expressed with u, u', v and v'.
How do you calculate this derivative?

But $$\vert\hspace{0.07cm}u(t)+i\cdot{}v(t)\vert = u^2(t) + v^2(t)$$

So, $$f'(t) = 2(uu' +vv')$$


EDIT : forgot SQRT, but Hurkyl got it !

 

[Hurkyl]

That won't work at all. In particular, $f = \sqrt{a^2}$ is incorrect and $df/da$ does not exist.


The most straightforward way to compute this derivative is to simply write out the function f. You recall that $|x + iy| = \sqrt{x^2 + y^2}$, right? Apply the definition of modulus, and you should get something you could do back in calc I.