# Differential calculus

## Limits

$$\lim_{x \rightarrow a}f(x)$$

$L^-$ - limit from below

$L^+$ - limit from above

$\lim_{x \to a} f(x)$ - limit of a point  

- Limit exists if $L^-=L^+$
- If limit exists, point does not.

Limits can be solved using normal techniques (if div 0, factorise)

## Limit theorems

1. For constant function $f(x)=k$, $\lim_{x \rightarrow a} f(x) = k$
2. $\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G$
3. $\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G$
4. ${\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0$

Corollary: $\lim_{x \rightarrow a} c \times f(x)=cF$ where $c=$ constant

## Solving limits for $x\rightarrow\infty$

Factorise so that all values of $x$ are in denominators.

e.g.

$$\lim_{x \rightarrow \infty}{{2x+3} \over {x-2}}={{2+{3 \over x}} \over {1-{2 \over x}}}={2 \over 1} = 2$$


## Continuous functions

A function is continuous if $L^-=L^+=f(x)$ for all values of $x$.

## Gradients of secants and tangents

Secant (chord) - line joining two points on curve

Tangent - line that intersects curve at one point

given $P(x,y) \quad Q(x+\delta x, y + \delta y)$:
gradient of chord joining $P$ and $Q$ is ${m_{PQ}}={\operatorname{rise} \over \operatorname{run}} = {\delta y \over \delta x}$

As $Q \rightarrow P, \delta x \rightarrow 0$. Chord becomes tangent (two infinitesimal points are equal).

Can also be used with functions, where $h=\delta x$.

## First principles derivative

$$f^\prime(x) = \lim_{\delta x \rightarrow 0}{\delta y \over \delta x}={dy \over dx}$$

$$m_{\tan}=\lim_{h \rightarrow 0}f^\prime(x)$$



$$m_{\vec{PQ}}=f^\prime(x)$$

first principles derivative:
$${m_{\text{tangent at }P} =\lim_{h \rightarrow 0}}{{f(x+h)-f(x)}\over h}$$

## Gradient at a point

Given point $P(a, b)$ and function $f(x)$, the gradient is $f^\prime(a)$


## Derivatives of $x^n$

$${d(ax^n) \over dx}=anx^{n-1}$$

If $x=$ constant, derivative is $0$

If $y=ax^n$, derivative is $a\times nx^{n-1}$

If $f(x)={1 \over x}=x^{-1}, \quad f^\prime(x)=-1x^{-2}={-1 \over x^2}$

If $f(x)=^5\sqrt{x}=x^{1 \over 5}, \quad f^\prime(x)={1 \over 5}x^{-4/5}={1 \over 5 \times ^5\sqrt{x^4}}$

If $f(x)=(x-b)^2, \quad f^\prime(x)=2(x-b)$

$$f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}$$

## Derivatives of $u \pm v$

$${dy \over dx}={du \over dx} \pm {dv \over dx}$$
where $u$ and $v$ are functions of $x$

## Euler's number as a limit

$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$

## Chain rule for $(f\circ g)$

If $f(x) = h(g(x)) = (h \circ g)(x)$:

$$f^\prime(x) = h^\prime(g(x)) \cdot g^\prime(x)$$

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.

e.g. $y=(x^2+5)^7$ - Cannot reasonably expand  
Let $u-x^2+5$ (inner expression)  
${du \over dx} = 2x$  
$y=u^7$  
${dy \over du} = 7u^6$  

## Product rule for $y=uv$

$${dy \over dx} = u{dv \over dx} + v{du \over dx}$$

## Quotient rule for $y={u \over v}$

$${dy \over dx} = {{v{du \over dx} - u{dv \over dx}} \over v^2}$$

$$f^\prime(x)={{v(x)u^\prime(x)-u(x)v^\prime(x)} \over [v(x)]^2}$$

## Logarithms

$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$

Wikipedia:

> the logarithm of a given number $x$ is the exponent to which another fixed number, the base $b$, must be raised, to produce that number $x$

### Logarithmic identities

$\log_b (xy)=\log_b x + \log_b y$  
$\log_b x^n = n \log_b x$  
$\log_b y^{x^n} = x^n \log_b y$

### Index identities

$b^{m+n}=b^m \cdot b^n$  
$(b^m)^n=b^{m \cdot n}$  
$(b \cdot c)^n = b^n \cdot c^n$  
${a^m \div a^n} = {a^{m-n}}$

### $e$ as a logarithm

$$\operatorname{if} y=e^x, \quad \operatorname{then} x=\log_e y$$
$$\ln x = \log_e x$$

### Differentiating logarithms
$${d(\log_e x)\over dx} = x^{-1} = {1 \over x}$$

## Derivative rules

| $f(x)$ | $f^\prime(x)$ |
| ------ | ------------- |
| $\sin x$ | $\cos x$ |
| $\sin ax$ | $a\cos ax$ |
| $\cos x$ | $-\sin x$ |
| $\cos ax$ | $-a \sin ax$ |
| $\tan f(x)$ | $f^2(x) \sec^2f(x)$ |
| $e^x$ | $e^x$ |
| $e^{ax}$ | $ae^{ax}$ |
| $ax^{nx}$ | $an \cdot e^{nx}$ |
| $\log_e x$ | $1 \over x$ |
| $\log_e {ax}$ | $1 \over x$ |
| $\log_e f(x)$ | $f^\prime (x) \over f(x)$ |
| $\sin(f(x))$ | $f^\prime(x) \cdot \cos(f(x))$ |
| $\sin^{-1} x$ | $1 \over {\sqrt{1-x^2}}$ |
| $\cos^{-1} x$ | $-1 \over {sqrt{1-x^2}}$ |
| $\tan^{-1} x$ | $1 \over {1 + x^2}$ |

## Reciprocal derivatives

$${1 \over {dy \over dx}} = {dx \over dy}$$

## Differentiating $x=f(y)$

Find $dx \over dy$. Then ${dx \over dy} = {1 \over {dy \over dx}} \implies {dy \over dx} = {1 \over {dx \over dy}}$.

$${dy \over dx} = {1 \over {dx \over dy}}$$

## Second derivative

$$f(x) \longrightarrow f^\prime (x) \longrightarrow f^{\prime\prime}(x)$$

$$\therefore y \longrightarrow {dy \over dx} \longrightarrow {d({dy \over dx}) \over dx} \longrightarrow {d^2 y \over dx^2}$$

Order of polynomial $n$th derivative decrements each time the derivative is taken

### Points of Inflection

*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

![](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:

$${dp \over dx} = {dq \over dx} \quad \text{and} \quad {dp \over dy} = {dq \over dy}$$

## Integration

$$\int f(x) \cdot dx = F(x) + c \quad \text{where } F^\prime(x) = f(x)$$

$$\int x^n \cdot dx = {x^{n+1} \over n+1} + c$$

- area enclosed by curves
- $+c$ should be shown on each step without $\int$

### Integral laws

$\int f(x) + g(x) dx = \int f(x) dx + \int g(x) dx$  
$\int k f(x) dx = k \int f(x) dx$  

| $f(x)$                          | $\int f(x) \cdot dx$         |
| ------------------------------- | ---------------------------- |
| $k$ (constant) | $kx + c$ |
| $x^n$ | ${x^{n+1} \over {n+1}} + c$ |
| $a x^{-n}$ | $a \cdot \log_e x + c$ |
| ${1 \over {ax+b}}$ | ${1 \over a} \log_e (ax+b) + c$ |
| $(ax+b)^n$ | ${1 \over {a(n+1)}}(ax+b)^{n-1} + c$ |
| $e^{kx}$ | ${1 \over k} e^{kx} + c$ |
| $e^k$ | $e^kx + c$ |
| $\sin kx$ | $-{1 \over k} \cos (kx) + c$ |
| $\cos kx$ | ${1 \over k} \sin (kx) + c$ |
| $\sec^2 kx$ | ${1 \over k} \tan(kx) + c$ |
| $1 \over \sqrt{a^2-x^2}$ | $\sin^{-1} {x \over a} + c \>\vert\> a>0$ |
| $-1 \over \sqrt{a^2-x^2}$ | $\cos^{-1} {x \over a} + c \>\vert\> a>0$ |
| $a \over {a^2-x^2}$ | $\tan^{-1} {x \over a} + c$ |
| ${f^\prime (x)} \over {f(x)}$ | $\log_e f(x) + c$ |
| $g^\prime(x)\cdot f^\prime(g(x)$ | $f(g(x))$ (chain rule)|
| $f(x) \cdot g(x)$ | $\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx$ |

Note $\sin^{-1} {x \over a} + \cos^{-1} {x \over a}$ is constant for all $x \in (-a, a)$.

### Definite integrals

$$\int_a^b f(x) \cdot dx = [F(x)]_a^b=F(b)-F(a)$$

- Signed area enclosed by: $\> y=f(x), \quad y=0, \quad x=a, \quad x=b$.
- *Integrand* is $f$.
- $F(x)$ may be any integral, i.e. $c$ is inconsequential

### Integration by substitution

$$\int f(u) {du \over dx} \cdot dx = \int f(u) \cdot du$$

Note $f(u)$ must be one-to-one $\implies$ one $x$ value for each $y$ value

e.g. for $y=\int(2x+1)\sqrt{x+4} \cdot dx$:  
let $u=x+4$  
$\implies {du \over dx} = 1$  
$\implies x = u - 4$  
then $y=\int (2(u-4)+1)u^{1 \over 2} \cdot du$  
Solve as a normal integral

#### Definite integrals by substitution

For $\int^b_a f(x) {du \over dx} \cdot dx$, evaluate new $a$ and $b$ for $f(u) \cdot du$.

### Trigonometric integration

$$\sin^m x \cos^n x \cdot dx$$

**$m$ is odd:**  
$m=2k+1$ where $k \in \mathbb{Z}$  
$\implies \sin^{2k+1} x = (\sin^2 z)^k \sin x = (1 - \cos^2 x)^k \sin x$  
Substitute $u=\cos x$

**$n$ is odd:**  
$n=2k+1$ where $k \in \mathbb{Z}$  
$\implies \cos^{2k+1} x = (\cos^2 x)^k \cos x = (1-\sin^2 x)^k \cos x$  
Subbstitute $u=\sin x$

**$m$ and $n$ are even:**  
Use identities:

- $\sin^2x={1 \over 2}(1-\cos 2x)$
- $\cos^2x={1 \over 2}(1+\cos 2x)$
- $\sin 2x = 2 \sin x \cos x


## Applications of antidifferentiation

- $x$-intercepts of $y=f(x)$ identify $x$-coordinates of stationary points on $y=F(x)$
- nature of stationary points is determined by sign of $y=f(x)$ on either side of its $x$-intercepts
- if $f(x)$ is a polynomial of degree $n$, then $F(x)$ has degree $n+1$

To find stationary points of a function, substitute $x$ value of given point into derivative. Solve for ${dy \over dx}=0$. Integrate to find original function.

## Rates

### Related rates

$${da \over db} \quad \text{(change in } a \text{ with respect to } b)$$

#### Gradient at a point on parametric curve

$${dy \over dx} = {{dy \over dt} \div {dx \over dt}} \> \vert \> {dx \over dt} \ne 0$$

$${d^2 \over dx^2} = {d(y^\prime) \over dx} = {{dy^\prime \over dt} \div {dx \over dt}} \> \vert \> y^\prime = {dy \over dx}$$

## 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