Maths - Product Rule | Olly Britton

Flashcards

If $y = uv$, what is $\frac{dy}{dx}$?

\[u\frac{dv}{dx} + v\frac{du}{dx}\]

If $f(x) = g(x)h(x)$, what is $f'(x)$?

\[g(x)h'(x) + h(x)g'(x)\]

How would you explain the product rule in English?

To find the derivative of two things multiplied together, add the products of the pairs of each function and its opposite’s derivative.

This diagram represents $h(x) = f(x)g(x)$. What’s the big square in the middle?

The output of the overall function.

This diagram represents $h(x) = f(x)g(x)$. What’s the two rectangles around the edge?

This represents the additional area gained when increasing the function by a small amount.

This diagram represents $h(x) = f(x)g(x)$. What represents the derivative?

The two slivers of area $df\times g$ and $dg \times f$.

This diagram represents $h(x) = f(x)g(x)$. Why isn’t $df\times dg$ accounted for in the product rule?

Since it’s an infinitesimal multiplied by an infinitesimal, it vanishes compared to the other derivatives.

\[f(x) = (x - 2)(x - 1)\]

What’s the derivative?

\[2x - 3\]

\[f(x) = (2x - 1)(3x + 7)\]

What’s the first step of the product rule?

\[3(2x - 1) + 2(3x + 7)\]

\[f(x) = e^x (2x - 1)\]

What’s the derivative?

\[e^x(2x + 1)\]