Further Maths - Leibnitz's Theorem
Pearson Edexcel Further Mathematics 2022
Flashcards
Building up the product rule
What is the derivative of
\[y = uv\]
?
\[u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}\]
What is the derivative of
\[\frac{\text{d}y}{\text{d}x} = u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}\]
?
\[u\frac{\text{d}^2v}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x} + v\frac{\text{d}^2u}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x}\]
Leibnitz’s theorem
What does Leibnitz’s theorem give a general formula for?
The $n$th derivative of the product of two functions.
What is the formula from Leibnitz’s theorem for the $n$th derivative of $y = uv$?
\[\frac{\text{d}^ny}{\text{d}x^n} = \sum^n _ {k = 0} \left(\begin{matrix} n \\\\ k \end{matrix}\right) \frac{\text{d}^ku}{\text{d}x^k} \frac{\text{d}^{n-k}v}{\text{d}x^{n-k}}\]
What goes in front of the derivatives in Leibnitz’s theorem?
Numbers from the $n$th row of Pascal’s triangle.
How could you more concisely write
\[\frac{\text{d}^n x}{\text{d}y^n} = m(m-1)(m-2)...(m-n+1)x^{m-n}\]
?
\[\frac{m!}{(m - n)!}x^{m-n}\]