Leibniz's Theorem

• Leibniz’s Theorem refers to various results attributed to the German mathematician and philosopher Gottfried Wilhelm Leibniz.

• In the context of Further Pure Mathematics 1, it deals mainly with the successive differentiation of a product of functions.

• Leibniz’s Rule, also known as the Product Rule, states that the derivative of a product of two functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.

• Mathematically, this can be expressed as: d/dx [f(x)g(x)] = f’(x)g(x) + f(x)g’(x).

• This theorem allows us to differentiate a product without having to simplify the product first.

• It is important to remember that this rule can be extended to the product of more than two functions.

• For the special case of the nth derivative of a product of two functions, Leibniz’s Rule takes a form similar to binomial expansion:

• The nth derivative of product of two functions u(x) and v(x) is given by: d^n(uv) / dx^n = Σ (from k=0 to n) C(n, k) (d^(n-k)u / dx^(n-k)) (d^k v / dx^k).

• In this formula, C(n, k) is the binomial coefficient, commonly known as “n choose k”.

• You need to be comfortable with working with this formula under various conditions, possibly including integration.

• Engage in plenty of practise questions with Leibniz’s Rule, making sure to cover a variety of question types to ensure a comprehensive grasp of the topic.

• Also remember, as with all rules in calculus, this rule must be used with caution. It only applies to functions that are continuous and differentiable on the given interval.