Partial Derivative wrt x of sin x y over cos (x + y)
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Theorem
- $\dfrac \partial {\partial x} \dfrac {\sin x y} {\map \cos {x + y} } = \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }$
Proof
\(\ds \frac \partial {\partial x} \frac {\sin x y} {\map \cos {x + y} }\) | \(=\) | \(\ds \frac {\map \cos {x + y} y \cos x y - \sin x y \paren {-\map \sin {x + y} } } {\map {\cos^2} {x + y} }\) | Quotient Rule for Derivatives, treating $y$ as a constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {y \map \cos {x + y} \cos x y + \map \sin {x + y} \sin x y} {\map {\cos^2} {x + y} }\) | simplifying |
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction: Exercise $1$