# Further Maths - Leibnitz's Theorem

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/leibnitz-theorem/ · Updated: 2021-11-24 · Tags: school, further-maths, year-2

## Flashcards
### 2021-11-24
##### 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 $$\frac{\text{d}y}{\text{d}x} = u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}$$??
$$
u\frac{\text{d}^2v}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x} + v\frac{\text{d}^2u}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x}
$$

##### What does Leibnitz's theorem give a general formula for??
The $n$th derivative of the product of two functions.

##### What is $\frac{\text{d}^0v}{\text{d}x^0}$??
$$
v
$$

##### What is the formula from Leibnitz's theorem for the $n$th derivative of $y = uv$??
$$
\frac{\text{d}^nx}{\text{d}y^n} = \sum^n_{k = 0} \left(\begin{matrix} n \\\\ k \end{matrix}\right) \frac{\text{d}^ku}{\text{d}x^k} \frac{\text{d}^{n-k}v}{\text{d}x^{n-k}}
$$

##### 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 $$\frac{\text{d}^n x}{\text{d}y^n} = m(m-1)(m-2)...(m-n+1)x^{m-n}$$??
$$
\frac{m!}{(m - n)!}x^{m-n}
$$

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