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If x is real, you can simplify f(x) to f(x) = h(x) + i * q(x), where h(x) and q(x) are real-valued functions, and i is the imaginary unit.
To derive f(x) into its real and imaginary parts separately:
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Real Part:
- Re(f(x)) = h(x)
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Imaginary Part:
- Im(f(x)) = q(x)
When differentiating f(x) with respect to x, and treating the real and imaginary parts separately:
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The derivative of f(x) is:
- d/dx [f(x)] = d/dx [h(x) + i * q(x)] = dh(x)/dx + i * dq(x)/dx
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Separating the derivative into real and imaginary parts:
- d/dx [f(x)] = dh(x)/dx + i * dq(x)/dx
Here, dh(x)/dx is the derivative of the real part h(x) with respect to x, and dq(x)/dx is the derivative of the imaginary part q(x) with respect to x.