Derivative of Cosine Integral Function

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Theorem

$\dfrac \d {\d x} \paren {\map \Ci x} = -\dfrac {\cos x} x$

where:

$\Ci$ denotes the cosine integral function
$x$ is a strictly positive real number.


Proof

\(\ds \frac \d {\d x} \paren {\map \Ci x}\) \(=\) \(\ds \frac \d {\d x} \paren {-\gamma - \ln x + \int_0^x \frac {1 - \cos t} t \rd t}\) Characterization of Cosine Integral Function
\(\ds \) \(=\) \(\ds -\frac 1 x + \frac 1 x - \frac {\cos x} x\) Derivative of Constant, Derivative of Natural Logarithm, Fundamental Theorem of Calculus: First Part (Corollary)
\(\ds \) \(=\) \(\ds -\frac {\cos x} x\)

$\blacksquare$


Also see