# j-james/math All my math notes, now in Markdown.

View the Project on GitHub j-james/math

# 3-3 Differentiation Rules

## Learning Targets

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

## Concepts / Definitions

### Derivative of a Constant

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

### Constant times a Function Rule

#### Solving for the rule

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

#### Definition

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

### The Sum and Difference Rule

#### Solving for the rule

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, $\frac{d}{dx}(u\pm v) = ?$

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

#### Definition

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, $$\frac{d}{dx} (u \pm v) = \frac{du}{dx} \pm \frac{dv}{dx}$$

### Power Rule for Derivatives

$f(x) = x^2$
x -1 0 1 2 3
f’(x) -2 0 2 4 6
$f'(x) = 2x$ $g(x) = x^3$
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
$g'(x) = 3x^2$ #### Definition

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

### The Product Rule

#### Solving

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

#### Definition

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

### The Quotient Rule

#### Definition

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”.

### Higher Order Derivatives

#### Second Derivative

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

#### Third Derivative

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

#### nth Derivative

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

### Examples

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

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

==>