## Chain rule for $(f\circ g)$
-$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
-$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
+If $f(x) = h(g(x)) = (h \circ g)(x)$:
-Function notation:
+$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
-$$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \mathbb{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$
+If $y=h(u)$ and $u=g(x)$:
+
+$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
Used with only one expression.
$y=u^7$
${dy \over du} = 7u^6$
-
## Product rule for $y=uv$
$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
Order of polynomial $n$th derivative decrements each time the derivative is taken
-### Maxima and minima
+### Points of Inflection
-- if $f^\prime (a) = 0$ and $f^{\prime\prime}(a) > 0$, then point $(a, f(a))$ is a local min (curve is concave up)
+*Stationary point* - point of zero gradient (i.e. $f^\prime(x)=0$)
+*Point of inflection* - point of maximum $|$gradient$|$ (i.e. $f^{\prime\prime} = 0$)
+- 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