From 4732ab1bfdc7cabb854018cb3424e02883c56a5e Mon Sep 17 00:00:00 2001
From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Wed, 3 Apr 2019 09:40:28 +1100
Subject: [PATCH] [spec] implicit differentiation

---
 spec/calculus.md | 16 ++++++++++++++--
 1 file changed, 14 insertions(+), 2 deletions(-)

diff --git a/spec/calculus.md b/spec/calculus.md
index a085f3b..c239084 100644
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -260,13 +260,25 @@ $${dy \over dx} = {{dy \over dt} \over {dx \over dt}} \> \vert \> {dx \over dt}
 
 $${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
 
-# Rational functions
+## Rational functions
 
 $$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
 
-## Addition of ordinates
+### Addition of ordinates
 
 - when two graphs have the same ordinate, $y$-coordinate is double the ordinate
 - when two graphs have opposite ordinates, $y$-coordinate is 0 i.e. ($x$-intercept)
 - when one of the ordinates is 0, the resulting ordinate is equal to the other ordinate
 
+
+## Implicit Differentiation
+
+On CAS: Action $\rightarrow$ Calculation $\rightarrow$ `impDiff(y^2+ax=5, x, y)`. Returns $y^\prime= \dots$.
+
+Used for differentiating circles etc.
+
+If $p$ and $q$ are expressions in $x$ and $y$ such that $p=q$, for all $x$ nd $y$, then:
+
+$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
+
+
-- 
2.47.1