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) | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {\cos x} x\) |
$\blacksquare$