+## 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 $\int$
| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
+### 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 $y=F(x)$
- 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
-
-
-## 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:
-
-$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$
-
-