Chain Rule

$\dfrac{df}{dx}=\dfrac{df}{du}\dfrac{du}{dx}$

The chain rule is helpful if we have an argument that is more than a single variable.

$f(x) = sin(ax)$

$\dfrac {d}{dx} f(x) = ?$

let u = ax, then $f(x) = sin(ax)$ and $f(u) = sin(u)$

$\dfrac {df} {du} = \dfrac {d}{du} sin(u) = cos(u)$

$\dfrac {du} {dx} = \dfrac {d}{dx} u = \dfrac {d}{dx} ax = a$

Putting these two pieces together we get

$\dfrac {df}{du} \dfrac {du} {dx} = cos(u) \: a = a \: cos(u) = a \: cos(ax)$

Try doing something similar to get the derivative of cos(ax). You should get

$\dfrac {d}{dx} cos(ax) = -a sin(ax)$ .

Here’s where it gets interesting for those who want to study Quantum Chemistry. From all the above, we can take the second derivative of sin(ax) and get the following:

$\dfrac {d^2}{dx^2} sin(ax) = -a^2 \: sin(ax)$

The function sin(ax) is an eigenfunction of $\dfrac {d^2}{dx^2}$ with an eigenvalue of $-a^2$ .

Be ready for more math like this as you journey into Quantum Chemistry.

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