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
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $25$.