Derivative of Composite Function/Examples/Exponential of a x^2

Example of Derivative of Composite Function

$\map {\dfrac \d {\d x} } {e^{a x^2} } = 2 a x e^{a x^2}$

Proof

Let $u = a x^2$.

Let $y = e^u$.

Thus we have:

$y = e^{a x^2}$

and so:

\(\ds \dfrac {\d y} {\d x}\)

\(=\)

\(\ds \dfrac {\d y} {\d u} \dfrac {\d u} {\d x}\)

Derivative of Composite Function

\(\ds \)

\(=\)

\(\ds e^u \cdot 2 a x\)

Power Rule for Derivatives, Derivative of Exponential Function

\(\ds \)

\(=\)

\(\ds 2 a x e^{a x^2}\)

simplification

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Exercises $\text {IX}$: $13$.