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3.2 The Derivative as a Function

Tangent line slope at \( x = a \)

\[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \]

\[ f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]

Figure: Graph of a function with a tangent line drawn at a point marked 'a' on the x-axis.

Instead of fixing \( x \) at \( a \), take the 2nd form and change \( a \) to \( x \):

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

This is a function of \( x \)

This gives the tangent line slope where this limit exists.

One condition needed:

Function must be continuous

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Example

\( f(x) = \sqrt{x} \) Domain: \( [0, \infty) \)

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

\[ = \lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h} \cdot \frac{\sqrt{x + h} + \sqrt{x}}{\sqrt{x + h} + \sqrt{x}} \quad \text{rationalize} \]

\[ = \lim_{h \to 0} \frac{(\sqrt{x + h})^2 - (\sqrt{x})^2}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \to 0} \frac{x + h - x}{h(\sqrt{x + h} + \sqrt{x})} \]

\[ = \lim_{h \to 0} \frac{h}{h(\sqrt{x + h} + \sqrt{x})} = \lim_{h \to 0} \frac{1}{\sqrt{x + h} + \sqrt{x}} = \frac{1}{\sqrt{x} + \sqrt{x}} = \frac{1}{2\sqrt{x}} \]

Figure: Graph of y equals square root of x, showing the curve and a tangent line at a point.

Slope = \( \frac{1}{2\sqrt{x}} \)

\( y = \sqrt{x} \)

Note \( f'(x) \) has domain \( (0, \infty) \)

Not same as \( f(x) \)

\( f'(0) \) DNE

\( f(x) \) is not differentiable at 0

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Graph of Derivative Functions

Graph of \( f'(x) = \frac{1}{2\sqrt{x}} \)

By observing the behavior of the derivative function, we can understand the slope of the tangent line to the original function:

High \( f'(x) \) \(\rightarrow\) tangent line very steep
Low \( f'(x) \) \(\rightarrow\) tangent line very shallow

Figure: Graph of f'(x) = 1/(2*sqrt(x)) showing a curve that starts high near the y-axis and approaches the x-axis.

Graphing \( f'(x) \) from \( f(x) \)

We can sometimes graph \( f'(x) \) by looking at the graph of \( f(x) \).

Example

Consider the graph of \( y = f(x) \):

Move left: slope becomes more negative.
At the vertex: tangent line is horizontal \(\rightarrow\) slope \(= f'(x) = 0\).
Move right: tangent line steeper \(\rightarrow f'(x) \) increases.

Figure: Two-part graph: top shows a parabola f(x); bottom shows its derivative f'(x) as a straight line through the origin.

As we move right on the graph of \( f(x) \), we observe a higher \( f'(x) \) value.

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Differentiability

\( f(x) \) is not differentiable (\( f'(x) \) DNE) if \( f(x) \):

Is not continuous at \( x \)
Has a vertical tangent line (\( f' \rightarrow \infty \) or DNE)
Is not smooth (sharp corner or cusp)

Figure: Graph illustrating a sharp corner at x=0 and a cusp at a positive x value.

Example: Sketching \( f'(x) \) from \( f(x) \)

Analyze the piecewise graph of \( f(x) \):

Discontinuous at \( x = 0, 5, 6 \)
Not smooth at \( x = 1, 3, 4 \)

Slopes of segments:

Slope = 1 for the first segment.
Slope = 0 at the peak.
Slope = 2 for the steep segment.
Slope = 1 for the final segment.

Figure: Piecewise function f(x) with various slopes and its derivative f'(x) graphed as constant horizontal segments.

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\( f'(x) \) tells us the rate of change of \( f(x) \)

For example,

on \( 0 < x < 1 \), \( f'(x) = 1 > 0 \) \( \rightarrow \) \( f(x) \) is increasing (positive rate)
on \( 4 < x < 5 \), \( f'(x) = -2 < 0 \) \( \rightarrow \) \( f(x) \) is decreasing (negative rate)

Example: Sketch graph of \( f(x) \) from the given graph of \( f'(x) \)

\( f'(x) \):

Slope = 1 at \( x = -2 \), then slope = 0 at \( x = -1 \) (horizontal tangent), then slope = -1 at \( x = 0 \)
Function not differentiable at \( x = 0 \) (possibly sharp corner or cusp)
Positive slope then flatten out at \( x = 1 \)

Graph of f'(x) with linear segments: from (-2, 1) to (0, -1) and from (0, 1) to (1, 0) . — Figure: Graph of \( f'(x) \) with linear segments: from \( (-2, 1) \) to \( (0, -1) \) and from \( (0, 1) \) to \( (1, 0) \).

\( f(x) \):

NOTE: this \( f(x) \) is not unique. Many possible \( f(x) \) to have the \( f'(x) \) above.

Two possible sketches of f(x) showing a cusp at x=0 and local extrema at x=-1 and x=1 . — Figure: Two possible sketches of \( f(x) \) showing a cusp at \( x=0 \) and local extrema at \( x=-1 \) and \( x=1 \).