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

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# 3-1 and 3-2 Definition of the Derivative and Differentiability

## Learning Targets

You will be able to

• Understand the meaning of the derivative
• Calculate stops and derivatives using the definition
• Graph $’f$ from $f$ and graph $f$ from $’f$
• Determine where a function is not differentiable and distinguish between the different types (corners, discontinuities, and vertical tangents)
• Calculate the numerical derivative (symmetric difference quotient)

## Concepts / Definitions

### Slope of a Curve at a Point

The slope of the curve $y = f(x)$ at the point $P(a,f(a))$ is the number $$m = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$$ provided the limit exists. ### Derivative at a Point

The derivative of the function $f$ at the point $x = a$ is $$f'(a) = \lim_{x\to a} \frac{f(x)-f(a)}{x-a}$$ provided the limit exists.

### The Derivative Function

What if we wanted to find the slope at any point, a functions of slopes, rather than slope at a particular point?

The derivative of the function $f$ with respect to the variable $x$ is the function $f’$ whose value at $x$ is
$$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ provided the limit exists. #### Other Notations for derivatives of $y = f(x)$

$$\frac{d}{dx}f(x) = \frac{df}{dx} = f'(x) = \frac{d}{dx}y = \frac{dy}{dx} = y'$$

### How $f’(a)$ fails to exist

#### Left derivative not equal to right derivative (a corner) #### A point where the function is discontinuous (discontinuity) #### Left and right slopes approach opposite $\infty$ (a cusp) ### Note: (todo)

Despite limits dealing with numbers, and $\infty$ not considered as one, mathmatical standard is to write the slope as $\infty$ in order to give more detail to the answer.

NOTE: Remember that for a derivative to exist at a point, the derivative (in other words, the limit of difference quotient) must exist as a number, and from the left and from the right must be the same. The limits for graphs II and III are $-\infty$ and $\infty$, respectively. It doesn’t exist for I and IV.

### Working Definitions

#### Differentiability Implies Continuity

If $f$ has a derivative at $x = a$, then $f$ is continuous at $x = a$.

#### Differentiability Implies Local Linearity

A differentiable function resembles its own tangent close (zoomed in) to some input value $a$.

#### Numerical Derivative

The numerical derivative of $f$ at $a$, which we will denote $nDeriv(f(x),a)$ is the number $$\frac{f(a+0.001)-f(a-0.001)}{0.002}$$

Then average the two below. Both must be really close to each other! $$f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$$ $$f'(a) = \lim_{h\to 0} \frac{f(a)=f(a-h)}{h}$$

### Graphing $f’$ from a graph of $f$ ### Determine $f$, $f’$, and $\frac{d}{dx}f’$ ==>