## 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 $y=uv$
$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$
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)
+![](graphics/second-derivatives.png)
+
+## 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:
+
+$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
## Antidifferentiation
## Integration
-$$\int f(x) dx = F(x) + c$$
+$$\int f(x) dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)$$
- area enclosed by curves
- $+c$ should be shown on each step without $\int$
| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+### Definite integrals
+
+$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)_{}$$
+
## Applications of antidifferentiation
- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \over {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$
+## Rational functions
+
+$$f(x) = {P(x) \over Q(x)} \quad \text{where } P, Q \text{ are polynomial functions}$$
+
+### 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