Product Rule for Derivatives/Examples/x times Cotangent of x
Jump to navigation
Jump to search
Examples of Use of Product Rule for Derivatives
- $\map {\dfrac \d {\d x} } {x \cot x} = \cot x - x \cosec^2 x$
Proof
Let $u = x$.
Let $v = \cot x$.
Thus we have:
- $y = u v$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds v \dfrac {\d u} {\d x} + u \dfrac {\d v} {\d x}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \cot x \cdot 1 + x \cdot \paren {-\cosec^2 x}\) | Power Rule for Derivatives, Derivative of Cotangent Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \cot x - x \cosec^2 x\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $16$.