Further Maths - Leibnitz's Theorem

Created: November 24, 2021 | About these notes | View in context | Study these flashcards

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 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 $\frac{\text{d}^0v}{\text{d}x^0}$?
\[v\]
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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