deriv-proofs.md (2070B)
1 --- 2 title: "Proof of Derivative Properties" 3 date: 2017-10-04 4 tags: math calculus proof notes 5 categories: math 6 math: true 7 --- 8 9 ## derivation of the quotient rule 10 11 The quotient rule is used to take the derivative of a function with divided 12 expressions: 13 14 $$ 15 \left(\frac{u}{v}\right)' = \frac{vu' - uv'}{v^2} 16 $$ 17 18 It is possible to prove this rule by utilizing the definition of the 19 derivative; however, this is not nearly as elegant as the following simple 20 proofs which use other derivative properties instead. 21 22 ### product rule 23 24 $$\begin{align} 25 y & = \frac{u}{v} \\ 26 & = uv^{-1} \\ 27 y' & = v^{-1}u' + u(-v^{-2}v') \\ 28 & = \frac{u'}{v} - \frac{uv'}{v^2} \\ 29 & = \frac{v}{v}\cdot\frac{u'}{v} - \frac{uv'}{v^2} \\ 30 & = \frac{vu'}{v^2} - \frac{uv'}{v^2} \\ 31 & = \frac{vu' - uv'}{v^2} \\ 32 \end{align}$$ 33 34 ### logarithm 35 36 $$\begin{align} 37 y & = \frac{u}{v} \\ 38 \mathrm{ln} y & = \mathrm{ln} \frac{u}{v} \\ 39 & = \mathrm{ln} u - \mathrm{ln} v \\ 40 \frac{y'}{y} & = \frac{u'}{u} - \frac{v'}{v} \\ 41 & = \frac{v}{v}\frac{u'}{u} - \frac{u}{u}\frac{v'}{v} \\ 42 & = \frac{vu' - uv'}{uv} \\ 43 y' & = y\frac{vu' - uv'}{uv} \\ 44 & = \frac{u}{v}\frac{vu' - uv'}{uv} \\ 45 & = \frac{vu' - uv'}{v^2} \\ 46 \end{align}$$ 47 48 ## logarithmic proofs 49 50 As well as being used in the proof of the quotient rule, logarithms can also be 51 used to prove a couple of other derivative rules. 52 53 ### power rule 54 55 $$\begin{align} 56 y & = x^n \\ 57 \mathrm{ln} y & = \mathrm{ln} x^n \\ 58 & = n \mathrm{ln} x \\ 59 \frac{y'}{y} & = n \frac{1}{x} \\ 60 y' & = yn \frac{1}{x} \\ 61 & = n x^n x^{-1} \\ 62 & = nx^{n-1}\\ 63 \end{align}$$ 64 65 ### product rule 66 67 $$\begin{align} 68 y & = uv \\ 69 \mathrm{ln} y & = \mathrm{ln} uv \\ 70 & = \mathrm{ln} u + \mathrm{ln} v \\ 71 \frac{y'}{y} & = \frac{u'}{u} + \frac{v'}{v} \\ 72 y' & = y \frac{u'}{u} + \frac{v'}{v} \\ 73 & = uv \left(\frac{u'}{u} + \frac{v'}{v}\right) \\ 74 & = vu' + uv' \\ 75 \end{align}$$