+---
+geometry: margin=2cm
+columns: 2
+graphics: yes
+tables: yes
+author: Andrew Lorimer
+classoption: twocolumn
+---
+
# Differential calculus
## Limits
+ ## 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
$$y={x^{n+1} \over n+1} + c$$
## 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 $\int$
| $f(x)$ | $\int f(x) \cdot dx$ |
| ------------------------------- | ---------------------------- |
| $k$ (constant) | $kx + c$ |
-| $x^n$ | ${1 \over {n+1}}x^{n+1} + c$ |
+| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
| $a x^{-n}$ | $a \cdot \log_e x + c$ |
+| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
+| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
| $e^k$ | $e^kx + c$ |
| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |
-| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
-| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+ ### 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
+ ## 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
-