Product Rule for Derivatives/Examples/x times Exponential of x times Sine of x
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Examples of Use of Product Rule for Derivatives
- $\map {\dfrac \d {\d x} } {x e^x \sin x} = e^x \paren {\paren {1 + x} \sin x + x \cos x}$
Proof
Let $u = x$.
Let $v = e^x$.
Let $w = \sin x$.
Thus we have:
- $y = u v w$
and so:
\(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds u v \dfrac {\d w} {\d x} + u w \dfrac {\d v} {\d x} + v w \dfrac {\d u} {\d x}\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds x e^x \cos x + x \sin x e^x + e^x \sin x \cdot 1\) | Power Rule for Derivatives, Derivative of Sine Function, Derivative of Exponential Function | |||||||||||
\(\ds \) | \(=\) | \(\ds e^x \paren {\paren {1 + x} \sin x + x \cos x}\) | simplification |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $21$.