From 9491e04038cd0f5e14830fd89794529b09bdd295 Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Tue, 26 Mar 2019 11:32:36 +1100
Subject: [PATCH] [spec] double derivatives

---
 spec/calculus.md | 30 +++++++++++++++++++++++++-----
 1 file changed, 25 insertions(+), 5 deletions(-)

diff --git a/spec/calculus.md b/spec/calculus.md
index 6c6685a..ee88068 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -111,8 +111,6 @@ $y=u^7$
 ${dy \over du} = 7u^6$  
 
 
-$7u^6 \times$
-
 ## Product rule for $y=uv$
 
 $${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
@@ -135,12 +133,14 @@ Wikipedia:
 
 > the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$
 
-### Logarithmic identities  
+### Logarithmic identities
+
 $\log_b (xy)=\log_b x + \log_b y$  
 $\log_b x^n = n \log_b x$  
 $\log_b y^{x^n} = x^n \log_b y$
 
 ### Index identities
+
 $b^{m+n}=b^m \cdot b^n$  
 $(b^m)^n=b^{m \cdot n}$  
 $(b \cdot c)^n = b^n \cdot c^n$  
@@ -154,7 +154,7 @@ $$\ln x = \log_e x$$
 ### Differentiating logarithms
 $${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$
 
-## Solving $e^x$ etc
+## Derivative rules
 
 | $f(x)$ | $f^\prime(x)$ |xs
 | ------ | ------------- |
@@ -182,7 +182,27 @@ $${{dy \over dx} \over 1} = dx \over dy$$
 
 ## Differentiating $x=f(y)$
 
-Find $dx \over dy$. Then $dx \over dy = {1 \over {dy \over dx}} \therefore {dy \over dx} = {1 \over {dx \over dy}$
+Find $dx \over dy$. Then $dx \over dy = {1 \over {dy \over dx}} \therefore {dy \over dx} = {1 \over {dx \over dy}}$.
+
+$${dy \over dx} = {1 \over {dx \over dy}}$$
+
+## Second derivative
+
+$$f(x) \implies f^\prime (x) \implies f^{\prime\prime}(x)$$
+
+$$\therefore y \implies {dy \over dx} \implies {d({dy \over dx}) \over dx} \implies {d^2 y \over dx^2}$$
+
+Order of polynomial $n$th derivative decrements each time the derivative is taken
+
+### Maxima and minima
+
+- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
+
+- if $f^\prime (a) = 0$ and $f^{\prime\prime} (a) < 0$, then point $(a, f(a))$ is local max (curve is concave down)
+- if $f^{\prime\prime}(a) = 0$, then point $(a, f(a))$ is a point of inflection
+- - if also $f^\prime(a)=0$, then it is a stationary point of inflection
+
+*Point of inflection* - point of maximum gradient (either +ve or -ve)
 
 ## Antidifferentiation
 
-- 
2.47.1