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

Learning Targets

You will be able to

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.

Slope of a Curve at a point

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.

The Derivative Function

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 Corner

A point where the function is discontinuous (discontinuity)


Left and right slopes approach opposite $\infty$ (a cusp)

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 $f$

f' from f

Determine $f$, $f’$, and $\frac{d}{dx}f’$

Determine f, f', and f''