Practical:Product rule for differentiation

This article considers practical aspects of the product rule for differentiation: how is this rule used in actual computations?

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Statement to remember

The statement of the product rule for differentiation that we will be using is:

${\frac {d}{dx}}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$

where $f'(x)={\frac {d(f(x))}{dx}}$ and $g'(x)={\frac {d(g(x))}{dx}}$.

Procedure to apply the product rule for differentiation

The product rule for differentiation is useful as a technique for differentiating functions that are expressed in the form of products of simpler functions.

Most explicit procedure

The explicit procedure is outlined below:

1. Identify the two functions whose product is the given function. In other words, explicitly decompose the function as a product of two functions. We will here call the functions $f$ and $g$, though you may choose to give them different names.
2. Calculate the derivatives of $f$ and $g$ separately, on the side.
3. Plug into the product rule formula the expressions for the functions and their derivatives.
4. Simplify the expression thus obtained (this is optional in general, though it may be required in some contexts).

Here is an example of a differentiation problem where we use this explicit procedure:

Differentiate the function $p(x):=(x^{2}+1)(x^{3}+2)$ with respect to $x$

We proceed step by step:

1. Identify the two functions: Define $f(x)=x^{2}+1$ and $g(x)=x^{3}+2$. Then, $p(x)=f(x)g(x)$ by definition.
2. Calculate the derivatives: The derivative of $f(x)$ is $f'(x)=2x$ and the derivative of $g(x)$ is $g'(x)=3x^{2}$.
3. Plug into the product rule formula: We get $p'(x)=f'(x)g(x)+f(x)g'(x)=(2x)(x^{3}+2)+(x^{2}+1)(3x^{2})$.
4. Simplify the expression obtained: We get $p'(x)=2x^{4}+4x+3x^{4}+3x^{2}=5x^{4}+3x^{2}+4x$.

Here is another example of a differentiation problem where we use this explicit procedure:

Differentiate the function $p(x):=e^{x}\sin x$

1. Identify the two functions: Define $f(x)=e^{x}$ and $g(x)=\sin x$
2. Calculate the derivatives: The derivative of $f(x)$ is $f'(x)=e^{x}$ and the derivative of $g(x)$ is $g'(x)=\cos x$.
3. Plug into the product rule formula: We get $p'(x)=f'(x)g(x)+f(x)g'(x)=(e^{x})(\sin x)+(e^{x})(\cos x)$.
4. Simplify the expression obtained: The expression is already simplified. If we wish to collect terms, we can rewrite as $e^{x}(\sin x+\cos x)$.

More inline procedure using Leibniz notation

Although the explicit procedure above is fairly clear, Step (2) of the procedure can be a waste of time in the sense of having to do the derivative calculations separately. If you are more experienced with doing differentiation quickly, you can combine Steps (2) and (3) by calculating the derivatives while plugging into the formula, rather than doing the calculations separately prior to plugging into the formula. Further, we do not need to explicitly name the functions if we use the Leibniz notation to compute the derivatives inline.

The shorter procedure is outlined below:

1. Identify the two functions being multiplied (but you don't have to give them names).
2. Plug into the formula for the product rule, using the Leibniz notation for derivatives that have not yet been computed.
3. Compute derivatives and simplify

For instance, consider the problem:

Differentiate the function $p(x):=(2x^{3}+3x-\sin x)(x^{2}+\cos x-3)$

The procedure is:

1. Identify the two functions: The functions are $x\mapsto 2x^{3}+3x-\sin x$ and $x\mapsto x^{2}+\cos x-3$.
2. Plug into the formula for the product rule: We get:
   ${\frac {d(2x^{3}+3x-\sin x)}{dx}}(x^{2}+\cos x-3)+(2x^{3}+3x-\sin x){\frac {d(x^{2}+\cos x-3)}{dx}}$
3. Compute derivatives and simplify: we get:
   $(6x^{2}+3-\cos x)(x^{2}+\cos x-3)+(2x^{3}+3x-\sin x)(2x-\sin x)$. The expression can be expanded and simplified if desired.

Shortest inline procedure

If you are really experienced with doing derivatives in your head, you can shorten the procedure even further by combining Steps (2) and (3) in the previous procedure. The procedure has two steps:

1. Identify the two functions being multiplied (but you don't have to give them names).
2. Use the formula for the product rule, computing the derivatives of the functions while plugging them into the formula

For instance, going back to the example used in the beginning:

Differentiate the function $p(x):=(x^{2}+1)(x^{3}+2)$ with respect to $x$

This can be done quickly:

1. Identify the two functions: They are $x\mapsto x^{2}+1$ and $x\mapsto x^{3}+2$.
2. Use the formula for the product rule, computing the derivatives of the functions while plugging them into the formula: We get $(2x)(x^{3}+2)+(x^{2}+1)(3x^{2})=5x^{4}+3x^{2}+4x$.

Choosing between procedures

The procedures are not fundamentally different, but they differ in the degree of explicitness of the steps. Generally speaking, the following are recommended:

• If the functions being multiplied are fairly easy to differentiate mentally, use the shortest inline procedure -- this is fast and reliable.
• If the functions being multiplied are somewhat more difficult to differentiate, then choose between the other two more explicit procedures, based on whether you are more comfortable with writing large inline expressions or with doing separate work on the side.

Error types

Incorrect formula

A common mistake in differentiating products of functions is the freshman product rule, i.e., the false rule that the derivative of the product is the product of the derivatives. The good news is that, generally speaking, it is easy to avoid this rule once you have enough experience with the actual product rule.

Writing only one piece of the product rule

This is an error of the incomplete task form and is harder to avoid. What happens here is that you forget to write one of the two pieces being added for the product rule, so perhaps you end up doing:

${\frac {d}{dx}}[f(x)g(x)]=f'(x)g(x)\qquad {\mbox{WRONG! Forgot one half of the product rule}}$

or

${\frac {d}{dx}}[f(x)g(x)]=f(x)g'(x)\qquad {\mbox{WRONG! Forgot one half of the product rule}}$

A slight variant involves forgetting one of the factors being multiplied:

${\frac {d}{dx}}[f(x)g(x)]=f'(x)+f(x)g'(x)\qquad {\mbox{WRONG! Forgot one of the pieces being multiplied}}$

Why this error occurs: Usually, this error is common if you are trying to use the shortest inline procedure, i.e., differentiating the functions and applying the product rule simultaneously, and one of the functions being differentiated is rather tricky to differentiate, requiring a product rule or chain rule for differentiation in and of itself.

How to avoid this error:

• When the functions being differentiated are tricky to differentiate, use either the fully explicit procedure or the inline procedure with Leibniz notation. Do not try to simultaneously differentiate the pieces and use the product rule.
• After finishing a product rule problem, ask the following sanity check question: did I get a sum of two products after the application of the product rule? If the answer is no, then check your work.

How to remember the formula

Different versions of the formula

Recall that we stated the product rule as:

${\frac {d}{dx}}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$

The right side could be written in any of eight equivalent ways:

1. $f'(x)g(x)+f(x)g'(x)$
2. $g(x)f'(x)+f(x)g'(x)$
3. $g(x)f'(x)+g'(x)f(x)$
4. $f'(x)g(x)+g'(x)f(x)$
5. $f(x)g'(x)+f'(x)g(x)$
6. $g'(x)f(x)+f'(x)g(x)$
7. $g'(x)f(x)+g(x)f'(x)$
8. $f(x)g'(x)+g(x)f'(x)$

All of these are equivalent, so in some sense it does not matter which of the versions you choose to remember. However, the first version is somewhat preferable to the others for a number of reasons mentioned below:

• The order of multiplication is the same as in the expression being differentiated: the $f$-part is on the left and the $g$-part is on the right. This is true only of versions (1) and (5). This is particularly important when we consider generalization to non-commutative situations such as product rule for differentiation of cross product.
• The differentiation symbol (the prime) hops from left to right: Remembering the rule this way makes it easier to generalize to differentiating products of more than two functions (see product rule for differentiation#Statement for multiple functions).

Quick rationalizations for the formula

If you are a little shaky about the product rule for differentiation, how do you do a reality check on the formula? A quick, intuitive version of the proof of product rule for differentiation using chain rule for partial differentiation will help. Basically, what it says is that to determine how the product changes, we need to count the contributions of each factor being multiplied, keeping the other constant. The contribution of $f$ keeping $g$ constant is $f'(x)g(x)$ and the contribution of $g$ keeping $f$ constant is $f(x)g'(x)$.

Applying the product rule for differentiation to compute values at a point

Knowledge of the values (in the sense of numerical values) $\!f(x_{0}),g(x_{0}),f'(x_{0}),g'(x_{0})$ at a specific point $x_{0}$ is sufficient to compute the value of $(f\cdot g)'(x_{0})$.

For instance, suppose we are given that $f(1)=5,g(1)=11,f'(1)=4,g'(1)=13$, we obtain that $(f\cdot g)'(1)=f'(1)g(1)+f(1)g'(1)=4\cdot 11+5\cdot 13=44+65=109$.

Dealing with products of three or more functions

One way of handling a product of three or more functions is to treat it as a product of two functions, where the functions themselves are products of the original functions provided, and then use the most explicit procedure. Another is to directly use the version of the product rule for differentiation that handles three or more functions, which makes things straightforward enough to use the inline procedure.

Most explicit procedure applied to a product of three or more functions

This essentially applies the #Most explicit procedure provided in a preceding section, with additional notes associated with this particular situation.

1. Identify the two functions: Break it as a product of two pieces each of which in turn is either one of the original functions or a product of subsets of the original functions.
   • For instance, let's say we are looking at a product $p(x):=f_{1}(x)f_{2}(x)f_{3}(x)$. We can break it down as $p(x)=f(x)g(x)$ where $f(x):=f_{1}(x)f_{2}(x)$ and $g(x):=f_{3}(x)$. So in this case, $f$ itself is a product, and $g$ is not.
   • If we are looking at a product of four functions, we can make $f$ the product of the first two functions and $g$ the product of the last two functions. In this case, both $f$ and $g$ are products.
   • If we are looking at a product of five or more functions, then one of $f$ and $g$ would be a product of three functions, which makes this whole process would have to be called on that as well (i.e., a recursive invocation).
2. Calculate the derivatives of $f$ and $g$ separately. We may need to use the product rule for differentiation for one or both of the calculations.
3. Plug into the product rule formula the expressions for the functions and their derivatives.
4. Simplify the expression thus obtained (this is optional in general, though it may be required in some contexts).

Here's an example:

Differentiate the function $f(x):=xe^{x}\ln x\sin x$ with respect to $x$

Note that the letter $f$ is being used for the whole function here, so let's use the letters $g$ and $h$ for the respective factors.

1. Identify the two functions: We'll set $g(x):=xe^{x}$ and $h(x):=\ln x\sin x$.
2. Calculate the derivatives: Using the product rule, we get $g'(x)=e^{x}+xe^{x}$, and Failed to parse (unknown function "\sinx"): {\displaystyle h'(x) = \frac{1}{x} \sinx + \ln x \cos x} .
3. Plug into the product rule formula: We get $f'(x)=g'(x)h(x)+g(x)h'(x)=(e^{x}+xe^{x})(\ln x\sin x)+(xe^{x})\left({\frac {1}{x}}\sin x+\ln x\cos x\right)$
4. Simplify the expression thus obtained: We get $f'(x)=e^{x}\ln x\sin x+xe^{x}\ln x\sin x+e^{x}\sin x+xe^{x}\ln x\cos x$.