All my math notes, now in Markdown.

You will be able to

- Use rules of differentiation to calculate derivatives
- Constant times function
- Sum or Difference of functions
- Product or Quotient rule of functions

- Apply derivatives of derivatives
- Use derivatives to find rates of change

If $f$ is the function with constant value $c$, then \(\frac{df}{dx} = \frac{d}{dx}(c) = 0\)

If $u$ is a differentiable function of $x$ and $c$ is a constant, then for $y = cu$ $\frac{d}{dx}cu = ?$

If $u$ is a differentiable function of $x$ and $c$ is a constant, then \(\frac{d}{dx}cu = c\frac{du}{dx}\)

If $u$ and $v$ are differentiable functions of $x$, then their sum and difference are differentiable at every point where $u$ and $v$ are differentiable. At such points, $\frac{d}{dx}(u\pm v) = ?$

\(y' = \lim_{h\to 0} \frac{u(x+h)+v(x+h)-(ux+vx)}{h}\) \(y' = \lim_{h\to 0} \frac{u(x+h)-ux+v(x+h)-vx}{h}\) \(y' = \lim_{h\to 0} \frac{u(x+h)-ux}{h} + \lim_{h\to 0} \frac{v(x+h)-vx}{h}\) \(y' = u'(x) + v'(x)\)

If $u$ and $v$ are differentiable functions of $x$, then their sum and difference are differentiable at every point where $u$ and $v$ are differentiable. At such points, \(\frac{d}{dx} (u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}\)

x | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|

f’(x) | -2 | 0 | 2 | 4 | 6 |

x | -1 | 0 | 1 | 2 | 3 | ||||
---|---|---|---|---|---|---|---|---|---|

g’(x) | 3 | 0 | 3 | 12 | 27 | ||||

V | V | V | V | ||||||

-3 | 3 | 9 | 15 | ||||||

V | V | V | |||||||

6 | 6 | 6 |

If $f(x) = x^n$ is a power function, with non-zero real number $n$, then \(\frac{d}{dx}f(x) = nx^{n-1}\)

\(f'(x) = \lim_{h\to 0} \frac{(x+h)^n - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{(_nC_0x^nh^0 + _nC_1x^{n-1}h^1 + _nC_2x^{n-2}h^2 + ... + _nC_{n-1}x^1h^{n-1} + _nC_nx^0h^n) - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{(x^n + nx^{n-1}h^1 + _nC_2x^{n-2}h^2 + ... + nx^1h^{n-1} + h^n) - x^n}{h}\) \(f'(x) = \lim_{h\to 0} \frac{h(nx^{n-1} + _nC_2x^{n-2}h^1 + ... + nx^1h^{n-2} + h^{n-1})}{h}\) \(f'(x) = \lim_{h\to 0} (nx^{n-1} + _nC_2x^{n-2}h^1 + ... + nx^1h^{n-2} + h^{n-1})\) \(\)

The product of two differentiable functions $u$ and $v$ is differentiable, and \(\frac{d}{dx}(uv) = \frac{du}{dx}v+u\frac{dv}{dx}\)

At a point where $v \neq 0$, the quotient $y = \frac uv$ of two differentiable functions is differentiable, and \(\frac{d}{dx}(\frac uv) = \frac{\frac{du}{dx}v - u\frac{dv}{dx}}{v^2}\)

In words, derivative of the numerator times the denominator minus the numerator times derivative of the denominator, all divided by the denominator squared.

Slightly simplier, derivative of the top times the bottom

minus the top times derivative of the bottom,

all divided by the bottom squared.

A common catch phrase to remember this is “Low d high minus high d low all divided by bottom squared”.

\(y'' = \frac{d}{dx}y' = \frac{d}{dx} (\frac{dy}{dx}) = \frac{d^2y}{dx^2}\)

\(y''' = \frac{d}{dx}y'' = \frac{d}{dx}(\frac{dy'}{dx}) = \frac{d^3y}{dx^3}\)

\(y^{(n)} = \frac{d}{dx}y^{n-1} = \frac{d^ny}{dx^n}\)

Find an equation of the line tangent to the curve $y = \frac{x^2 - 1}{x^2 + 1}$ at $x = -2$

\(\frac{dy}{dx} = \frac{4x}{(x^2+1)^2}\) \(\frac{dy}{dx}\rvert_{x = -2} = \frac{4(-2)}{((-2)^2 + 1)^2}\) \(\frac{dy}{dx}\rvert_{x = -2} = -\frac{8}{25}\) \(y = \frac{(-2)^2-1}{(-2)^2+1}\) \(y = \frac 35\) \(z = -\frac{8}{25}(x+2) + \frac 35\)