Product Rule for Derivatives/Examples/2 a x times Exponential of a x^2
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Examples of Use of Product Rule for Derivatives
- $\map {\dfrac \d {\d x} } {2 a x e^{a x^2} } = 2 a e^{a x^2} \paren {1 + 2 a x^2}$
Proof
Let $u = 2 a x$.
Let $v = e^{a x^2}$.
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 | |||||||||||
\(\ds \) | \(=\) | \(\ds e^{a x^2} \cdot 2 a + 2 a x \cdot \paren {2 a x e^{a x^2} }\) | Power Rule for Derivatives, Derivative of $e^{a x^2}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 a e^{a x^2} \paren {1 + 2 a x^2}\) | simplification |
$\blacksquare$
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