Product rule for higher derivatives

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This article is about a differentiation rule, i.e., a rule for differentiating a function expressed in terms of other functions whose derivatives are known.
View other differentiation rules

Statement

Version type Statement
specific point, named functions This states that if f and g are n times differentiable functions at x = x_0, then the pointwise product f \cdot g is also n times differentiable at x = x_0, and we have:
\frac{d^n}{dx^n}[f(x)g(x)]|_{x = x_0} = \sum_{k=0}^n \binom{n}{k} f^{(n - k)}(x_0)g^{(k)}(x_0)
Here, f^{(n - k)} denotes the (n-k)^{th} derivative of f (with f^{(0)} = f, f^{(1)} = f', etc.), g^{(k)} denotes the k^{th} derivative of g, and \binom{n}{k} is the binomial coefficient. These are the same as the coefficients that appear in the expansion of \! (A + B)^n.
generic point, named functions, point notation If f and g are functions of one variable, the following holds wherever the right side makes sense:
\! \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}(x)g^{(k)}(x)
generic point, named functions, point-free notation If f and g are functions of one variable, the following holds wherever the right side makes sense:
\! (fg)^{(n)} = \sum_{k=0}^n \binom{n}{k} f^{(n-k)}g^{(k)}
Pure Leibniz notation Suppose u and v are both variables functionally dependent on x. Then \frac{d^n(uv)}{(dx)^n} = \sum_{k=0}^n \binom{n}{k} \frac{d^{n-k}u}{(dx)^{n-k}}\frac{d^kv}{(dx)^k}

One-sided version

There are analogues of each of the statements with one-sided derivatives. Fill this in later

Particular cases

Value of n Formula for \frac{d^n}{dx^n}[f(x)g(x)]
1 \! f'(x)g(x) + f(x)g'(x) (this is the usual product rule for differentiation).
2 \! f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x).
3 \! f'''(x)g(x) + 3f''(x)g'(x) + 3f'(x)g''(x) + g'''(x).
4 \! f^{(4)}(x)g(x) + 4f'''(x)g'(x) + 6f''(x)g''(x) + 4f'(x)g'''(x) + f(x)g^{(4)}(x)
5 \! f^{(5)}(x)g(x) + 5f^{(4)}(x) g'(x) + 10f'''(x)g''(x) + 10f''(x)g'''(x) + 5f(x)g^{(4)}(x) + g^{(5)}(x)

Related rules

Similar rules in single variable calculus

Similar rules in multivariable calculus

Similar rules in advanced mathematics

Significance

Qualitative and existential significance

Each of the versions has its own qualitative significance:

Version type Significance
specific point, named functions This tells us that if f and g are both n times differentiable at a point x_0, so is f \cdot g.
generic point, named functions, point notation This tells us that if f and g are both n times differentiable on an open interval, so is f \cdot g.
generic point, named functions, point-free notation This shows that the way that (f \cdot g)^{(n)} behaves is governed by the nature of the derivatives (up to the n^{th}) of f and g. In particular, if f^{(n)} and g^{(n)} are both continuous functions on an interval, so is (f \cdot g)^{(n)}.

Computational feasibility significance

Each of the versions has its own computational feasibility significance:

Version type Significance
specific point, named functions This tells us that knowing the values (in the sense of numerical values) of f,f',f'',\dots, f^{(n)} and g,g',g'',\dots,g^{(n)} at a point x_0 allows us to compute the value (f \cdot g)^{(n)}(x_0) by plugging into the formula and doing a bunch of multiplications and additions.
generic point, named functions This tells us that knowledge of the generic expressions for f,f',f'',\dots, f^{(n)} and g,g',g'',\dots,g^{(n)} allows us to compute the generic expression for (f \cdot g)^{(n)}.