Quotient Rule for Derivatives/Examples/x over Cosine of x

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Example of Use of Quotient Rule for Derivatives

$\map {\dfrac \d {\d x} } {\dfrac x {\cos x} } = \dfrac {\cos x + x \sin x} {\cos^2 x}$


Proof

\(\ds \map {\dfrac \d {\d x} } {\dfrac x {\cos x} }\) \(=\) \(\ds \dfrac {\cos x \map {\frac \d {\d x} } x - x \map {\frac \d {\d x} } {\cos x} } {\cos^2 x}\) Quotient Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {\cos x \cdot 1 - x \cdot \paren {-\sin x} } {\cos^2 x}\) Derivative of Cosine Function, Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac {\cos x + x \sin x} {\cos^2 x}\) simplification

$\blacksquare$


Sources