Product Rule
The product rule states that the derivative of two functions that are multiplied together is the first function multipled by the derivative of the second function, plus the derivative of the first function multiple by the second function. This can be remembered with the nemonic:
right d left plus left d right
The derivative is therefore given by:
\[ g(x)h(x)’ = g(x)\frac{dh}{dx} + h(x)\frac{dg}{dx} \]
or
\[ u’v + v’u \]
Examples:
\[ (4t^{2} - t)(t^3 - 8t^2 + 12)’ = \] \[ (4t^{2} - t)(3t^2 - 16t) + (t^3 - 8t^2 + 12)(8t - 1) = \] \[ 20t^4 - 132t^3 + 24t^2 + 96t - 12 \]
Proof
You can prove that \( (uv)’ = u’v + v’u \) by using the multiplication of these functions with the \( \frac{dy}{dx} \) formula. In the example below I use the functions \( f \) and \( g \). First we substitute into the change in \( x \) formula with the two functions multiplied together.
\[ \frac{f(x + \Delta x)g(x + \Delta x) - f(x)g(x)}{\Delta x} \]
We then add by \( f(x + \Delta x)g(x) \) and \( -f(x + \Delta x)g(x) \) which equals \( 0 \) but allows us to group terms together. This results in:
\[ \frac{f(x + \Delta x)g(x + \Delta x) -f(x + \Delta x)g(x) + f(x + \Delta x)g(x) - f(x)g(x)}{\Delta x} \]
Next, factorise the resulting terms.
\[ \frac{f(x + \Delta x)(g(x + \Delta x) - g(x))}{\Delta x} + \frac{ g(x)(f(x + \Delta x) - f(x))}{\Delta x} \]
Bring the first term of each fraction outside of the divisor to separate them.
\[ f(x + \Delta x)\frac{g(x + \Delta x) - g(x)}{\Delta x} + g(x)\frac{ f(x + \Delta x) - f(x)}{\Delta x} \]
We now have the derivative defined in four distinct units: \( f(x + \Delta x) \), \( \frac{(g(x + \Delta x) - g(x))}{\Delta x} \), \( g(x) \) and \( \frac{f(x + \Delta x) - f(x)}{\Delta x} \). This format is starting to look suspiciously like \( u’v + v’u \). The last piece of puzzle is to define individuals limits for each part of this expression and use the limit sum rule to combine them.
\[ \lim_{x \to 0} f(x + \Delta x) = f(x) \]
\[ \lim_{x \to 0} \frac{(g(x + \Delta x) - g(x))}{\Delta x} = g’(x) \]
\[ \lim_{x \to 0} g(x) = g(x) \]
\[ \lim_{x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} = f’(x) \]
So therefore, if we place the limits into the original long form expression
\[ \lim_{x \to 0} f(x + \Delta x) \lim_{x \to 0} \frac{g(x + \Delta x) - g(x)}{\Delta x} + \lim_{x \to 0} g(x) \lim_{x \to 0} \frac{ f(x + \Delta x) - f(x)}{\Delta x} \]
By substituting for equivalent expressions defined above we can see that
\[ = f(x)g’(x) + g(x)f’(x) \]
Which is the same as \( (f(x)g(x))’ \) thus proving the product rule.
Multiple Products
This generalises further to finding the derivate of expressions with more than two functions multipled together so that you take the derivate of each individual function multiplied by the other functions and added together for each combination of functions, in the form:
\[ g(x)h(x)f(x)’ = g(x)’h(x)f(x) + g(x)h(x)’f(x) + g(x)h(x)f(x)’ \]
Visualising the Product Rule
You can visualise the product rule using simple geometric shapes. For example, in the case of two functions being multiplied together you can imagine this as a rectangle who’s length is given by the first function and who’s width is given by the second function. The example below shows a box, the length of which is given by the function sin(x) and the height of which is given by x^2.
┏━━sin(x)━━┓
┏━━━━━━━━━━┓ ┓
┃ ┃ ┃
┃ ┃ x^2
┃ ┃ ┃
┗━━━━━━━━━━┛ ┛
If you were to increase x slightly you would get 3 new pieces of area: A, B and C.
┏━━━━━━━━━━┓ ┏━┓
┃ ┃ ┃ ┃
┃ ┃ ┃A┃
┃ ┃ ┃ ┃
┗━━━━━━━━━━┛ ┗━┛
┏━━━━━━━━━━┓ ┏━┓
┃ B ┃ ┃C┃
┗━━━━━━━━━━┛ ┗━┛
The areas of these different sections are:
\[ A = \Delta sin(x)x^2 \]
\[ B = sin(x)\Delta x^2 \]
\[ C = \Delta sin(x) \Delta x^2 \]
We can ignore C because the change of a Δ value raised to a power is negligible. That means the whole change of this expression over a tiny Δx is the area of A plus the area of B which is the same as product rule outlined above. So the solution would be:
\[ cos(x)x^2 + sin(x)2x \]
This geometric intuition can be generalised to three dimensions to implicate what happens with a higher number of functions. In that case you would end up with three cuboids that describe the derivative of the function.