[spec] rates

diff --git a/spec/calculus.md b/spec/calculus.md
index ee880685f6e7af164317681ea73ce5ec9b557e45..c59c4b77e306241793268f6d18b4bd26c6efdf7e 100644 (file)
--- a/spec/calculus.md
+++ b/spec/calculus.md
@@ -245,3 +245,15 @@ $\int k f(x) dx = k \int f(x) dx$
 
 To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
 
 
 To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.
 

+## Rates
+
+### Related rates
+
+$${da \over db} \quad \text{change in } a \text{ with respect to } b$$
+
+#### Gradient at a point on parametric curve
+
+$${dy \over dx} = {{dy \over dt} \over {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$
+
+$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
+