## 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 $f(x) = h(g(x)) = (h \circ g)(x)$:
+
+$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$
-Function notation:
+If $y=h(u)$ and $u=g(x)$:
-$$(f\circ g)^\prime(x)=f^\prime(g(x))g^\prime(x),\quad \mathbb{where}\hspace{0.3em} (f\circ g)(x)=f(g(x))$$
+$${dy \over dx} = {dy \over du} \cdot {du \over dx}$$
+$${d((ax+b)^n) \over dx} = {d(ax+b) \over dx} \cdot n \cdot (ax+b)^{n-1}$$
Used with only one expression.
$y=u^7$
${dy \over du} = 7u^6$
-
## Product rule for $y=uv$
$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
### Points of Inflection
-*Point of inflection* - point of maximum gradient (either +ve or -ve). Occurs where $f^{\prime\prime} = 0$
+*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)