Basic Derivative Information: Summary and Review

This lesson reviews and summarizes basic derivative information.

Derivatives Give Slope Information

The derivative of a function $\,f\,$ is a (new) function that gives the slopes of the tangent lines to the graph of $\,f\,.$

Derivatives give slope information.

Remember that (when they exist) tangent lines capture the direction you're moving as you travel along a curve .

Prime Notation for the Derivative

Using prime notation, the derivative of $\,f\,$ is named $\,f'\,.$ The function $\,f'\,$ is read aloud as ‘$\,f\,$ prime’.

Notice that $\,f\,$ is a function, and $\,f'\,$ is also a function. Usually (but not always!) the function $\,f'\,$ is different from the function $\,f\,.$

The function $\,f'\,$ gives the slopes of the tangent lines to the graph of $\,f\,.$

Here is a simple example to illustrate the derivative concept and prime notation:

a simple derivative example

$f(x) = (x-1)^2 + 2$
Function rule that generates the graph; a parabola with vertex $\,(1,2)$

$f(1) = (1-1)^2 + 2 = 2$
The point $\,(1,2)\,$ lies on the graph of $\,f$

Derivative information at a point:

$$f'(1) = 0$$

Labeling the parts:

$$ \overbrace{f'}^{\text{slope of tangent line}} \underbrace{(1)}_{\text{when $x$ is $1$}}\ \ \overbrace{= 0}^{\text{is zero}} $$

The slope of the tangent line (shown in green), when $\,x = 1\,,$ is zero (horizontal line).
The function $\,f'\,$ gives slope information!

As you travel along the curve (going from left to right), at the instant that you pass through the point $\,(1,2)\,,$ you're moving horizontally!

The $\,\frac{d}{dx}\,$ Operator for Differentiation

The $\,\frac{d}{dx}\,$ operator is an instruction to ‘take the derivative with respect to $\,x\,$ of whatever comes next’. You can read ‘$\,\frac{d}{dx}\,$’ as ‘dee dee x’ .

For example, $\,\frac{d}{dx}(x^2)\,$ denotes the derivative with respect to $\,x\,$ of $\,x^2\,.$

You can read  ‘$\,\frac{d}{dx}(x^2)\,$’   as   ‘dee dee x   of   x squared’ .

Similarly, $\,\frac{d}{dt}(t^2)\,$ denotes the derivative with respect to $\,t\,$ of $\,t^2\,.$

The $\,\frac d{dx}\,$ operator is particularly useful when differentiating a function that has not been given a name.

If a function has a name (like $\,f\,$) then the prime notation for the derivative ($\,f'\,$) is usually easier.

If you're differentiating a function that uses $\,x\,$ as its input variable, then use $\,\frac{d}{dx}\,$ to request its derivative. If a function uses $\,t\,$ to represent the input, then use $\,\frac{d}{dt}\,$ to request its derivative (and so on).

If there is no confusion about the function that $\,\frac{d}{dx}\,$ is acting on, then the parentheses that hold the function to be differentiated may be dropped.

For example, you can write $\,\frac d{dx} x^2\,$ (no parentheses) instead of $\,\frac d{dx}(x^2)\,$ (with parentheses).

Equivalent Characterization of Differentiability

If a function $\,f\,$ is differentiable at $\,x\,,$ then there is a non-vertical tangent line to the graph of $\,f\,$ at the point $\,\big(x,f(x)\bigr)\,.$

Also, if there is a non-vertical tangent line to the graph of $\,f\,$ at the point $\,\big(x,f(x)\bigr)\,,$ then the function $\,f\,$ is differentiable at $\,x\,.$

Thus:

$f\,$ is differentiable at $\,x$

is equivalent to

there is a non-vertical tangent line to the graph of $\,f\,$ at the point $\,\bigl(x,f(x)\bigr)$

$f'(x)\,$ is a Number That Gives the Slope of a Tangent Line

The function $\,f'\,,$ evaluated at $\,x\,,$ is denoted by $\,f'(x)\,.$

$f'(x)\,$ is read aloud as   ‘$f\,$ prime of $\,x\,$’.

$f'(x)\,$ is an expression. When $\,f'(x)\,$ exists, then it is a number.

The number $\,f'(x)\,$ is the derivative of $\,f\,$ at $\,x\,.$

The number $\,f'(x)\,$ gives the slope of the (non-vertical) tangent line to the graph of $\,f\,$ at the point $\,\bigl(x,f(x)\bigr)\,.$

$f'\,$ is a Function; $\,f'(x)\,$ is a Number

Be careful:

If You're Worried About Differentiability at an Isolated Point ...

... then you're beyond the level of this Calculus course. (Think about pursuing a career in mathematics!)

When we talk about a function being differentiable at a point, we're assuming it's not an isolated point.

If you don't know what this is all about—just laugh and move on!

Conditions for a Function $\,f\,$ to be Differentiable at $\,x$

For $\,f\,$ to be differentiable at $\,x\,,$ all the following requirements must be met:

Every Important Calculus Idea is Defined in Terms of a Limit

Every important calculus idea is defined in terms of a limit. As an example, the derivative is defined in terms of a limit—keep reading!

The Definition of Derivative

difference quotients, h positive

Here is the definition of $\,f'(x)\,.$ (Note that it's defined in terms of a limit!) As long as the limit exists, then:

$$f'(x) := \lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h}$$

Recall that ‘$\,:=\,$’ means ‘equals, by definition’.

Understanding the Expression that Appears in the Definition of Derivative

difference quotients, h negative

The expression

$$\frac{f(x+h) - f(x)}{h}$$

gives the slope of the line between $\,\bigl(x,f(x)\bigr)\,$ and a (nearby) point $\,\bigl(x+h,f(x+h)\bigr)\,.$

Checking that this is indeed true:

$$ \begin{align} &\text{slope of line through}\cr &\bigl(x,f(x)\bigr) \text{ and } \bigl(x+h\ ,\ f(x+h)\bigr) \cr\cr &\qquad = \frac{y_2 - y_1}{x_2 - x_1}\cr\cr &\qquad = \frac{f(x+h)-f(x)}{(x+h)-x}\cr\cr &\qquad = \frac{f(x+h)-f(x)}h \end{align} $$

(Notice that $\,x\,$ cancels out in the denominator.)

By taking the limit as $\,h\,$ approaches $\,0\,,$ the point $\,\bigl(x+h,f(x+h)\bigr)\,$ is ‘slid’ closer and closer to the original point $\,\bigl(x,f(x)\bigr)\,.$ Notice that as $\,h\,$ gets closer to zero, $\,x + h\,$ gets closer to $\,x\,.$

The definition of derivative solves the essential problem in finding the slope of a tangent line:  usually, we only know one point on a tangent line. But, we need two points to compute a slope!

How is this problem solved? First, get the point where we want the tangent line:  this is the point $\,(x,f(x))\,.$ Then, get a nearby point:  this is the point $\,(x+h\,,\,f(x+h))\,.$  Slide the nearby point closer and closer by making $\,h\,$ smaller and smaller!

Equivalent Characterizations of ‘$\,f\,$ is Differentiable at $\,x\,$’

In summary, the following are equivalent for a number $\,x\,$ in the domain of $\,f\,$:

What if the Limit that Appears in the Definition of Derivative Doesn't Exist?

If the limit

$$\lim_{h\rightarrow 0} \frac{f(x+h) - f(x)}{h} $$

doesn't exist, then we say that ‘$\,f\,$ is not differentiable at $\,x\,$’   or   ‘$\,f'(x)\,$ does not exist’ . The phrase ‘does not exist’ is often abbreviated as DNE.

Equivalent Characterizations of ‘$\,f\,$ is Not Differentiable at $\,x\,$’

The following are equivalent for a number $\,x\,$ in the domain of $\,f\,$:

The Definition of Derivative is Tedious to Use

The definition of derivative is tedious/difficult to use, particularly as the function $\,f\,$ increases in complexity.

There must be a better way to find $\,f'(x)\,,$ rather than having to use the (limit) definition of derivative! There is! Start learning the Basic Differentiation Shortcuts here.

Concept Practice