From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 8 Aug 2018 05:08:25 +0000 (+1000)
Subject: differentiation
X-Git-Tag: yr11~73
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/a225a5d30844ad7f4d65c437aa9229fab409b84f

differentiation
---

diff --git a/spec/calculus.md b/spec/calculus.md
index 28ca344..d1a981c 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -33,7 +33,39 @@ e.g.
 $$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$
 
 
-
 ## Continuous functions
 
 A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.
+
+## Gradients of secants and tangents
+
+Secant (chord) - line joining two points on curve
+
+Tangent - line that intersects curve at one point
+
+given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
+gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$
+
+As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).
+
+Can also be used with functions, where $h=\delta x$.
+
+## First principles derivative
+
+$$\lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx} = f^\prime(x)$$
+
+$$m_{\operatorname{tangent}}=\lim_{h \rightarrow 0}f^\prime(x)$$
+
+
+
+$$m_{\operatorname{chord PQ}}=f^\prime(x)$$
+
+first principles derivative:
+$${m_{\operatorname{tangent at P}} =\lim_{h \rigzhtarrow 0}}{{f(x+h)-f(x)}\over h}$$
+
+
+
+
+## Euler's number as a limit
+
+$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$