2.8 The Derivative as a Function

The Derivative as a Function

• The derivative of a function $f$ at a fixed number $a$ has been previously considered.

• Now, the focus shifts to letting the number $a$ vary.

• By replacing $a$ with a variable $x$, a new function is obtained, denoted as $f'(x)$.

Definition of the Derivative Function

• For any number $x$ where the following limit exists, we assign to $x$ the number $f'(x)$.

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

• This defines a new function $f'$, called the derivative of $f$.

• The value of $f'(x)$ is interpreted geometrically as the slope of the tangent line to the graph of $f$ at the point $(x, f(x))$.

• $f'$ is "derived" from $f$ through a limiting operation.

• The domain of $f'$ may be smaller than the domain of $f$.

Example 1: Sketching the Graph of a Derivative

• To estimate the value of the derivative at any value of $x$, draw the tangent at the point $(x, f(x))$ and estimate its slope.

• For instance, at $x = 5$, draw the tangent at point $P$ and estimate its slope.

• Plot the point on the graph of $f'$ directly beneath $P$.

• Repeat this procedure at several points to obtain the graph of $f'$.

• Tangents at points $A$, $B$, and $C$ are horizontal, indicating the derivative is 0 there.

• The graph of $f'$ crosses the x-axis at points directly beneath $A$, $B$, and $C$.

• Between $A$ and $B$, tangents have positive slope, so $f'$ is positive.

• Between $B$ and $C$, tangents have negative slope, so $f'$ is negative.

• When $x$ is close to 0, $f'$ is also close to 0, so $f'(x)$ is very large; this corresponds to the steep tangent lines near $(0, 0)$

• When $x$ is large, $f'(x)$ is very small, corresponding to flatter tangent lines at the far right of the graph of $f$, and the horizontal asymptote of the graph of $f'$.

Other Notations for Derivatives

• If $y = f(x)$, alternative notations for the derivative are:

  • $f'(x) = y' = \frac{dy}{dx} = Df(x) = D_x f(x)$

• The symbols $D$ and $\frac{d}{dx}$ are differentiation operators.

• Leibniz notation: $\frac{dy}{dx}$ is a synonym for $f'(x)$, not a ratio for the time being.

• Definition of derivative in Leibniz notation:

  • $\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$

• To indicate the value of a derivative in Leibniz notation at a specific number $a$:

  • $\frac{dy}{dx} \Big|_{x=a}$

• This is a synonym for $f'(a)$.

• Definition: A function $f$ is differentiable at $a$ if $f'(a)$ exists.

• It is differentiable on an open interval $(a, b)$ if it is differentiable at every number in the interval.

Example 5: Determining Where a Function Is Differentiable

• Consider the function $f(x) = |x|$

• For x > 0, $|x| = x$, and for x < 0, $|x| = -x$

• For x > 0, we have

  • $f'(x) = \lim<em>{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim</em>{h \to 0} \frac{(x+h) - x}{h} = \lim_{h \to 0} \frac{h}{h} = 1$

• So, $f$ is differentiable for any x > 0.

• Similarly, for x < 0, we have

  • $f'(x) = \lim<em>{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim</em>{h \to 0} \frac{-(x+h) - (-x)}{h} = \lim_{h \to 0} \frac{-h}{h} = -1$

• So, $f$ is differentiable for any x < 0.

• For $x = 0$, we investigate:

  • $\lim<em>{h \to 0} \frac{f(0+h) - f(0)}{h} = \lim</em>{h \to 0} \frac{|0+h| - |0|}{h} = \lim_{h \to 0} \frac{|h|}{h}$

• Let's compute the left and right limits separately:

  • $\lim<em>{h \to 0^+} \frac{|h|}{h} = \lim</em>{h \to 0^+} \frac{h}{h} = 1$

  • $\lim<em>{h \to 0^-} \frac{|h|}{h} = \lim</em>{h \to 0^-} \frac{-h}{h} = -1$

• Since these limits are different, $f'(0)$ does not exist.

• Thus, $f$ is differentiable at all $x$ except 0.

• A formula for $f'(x)$ is given by:

• The fact that $f'(0)$ does not exist geometrically means that the curve does not have a tangent line at $(0, 0)$.

Relationship Between Differentiability and Continuity

• Theorem: If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

• Note: The converse is false, meaning there are functions that are continuous but not differentiable.

How Can a Function Fail to Be Differentiable?

• A function can fail to be differentiable if:

  • The graph of $f$ has a "corner" or "kink".

  • $f$ is not continuous at $a$.

  • The curve has a vertical tangent line when $x = a$.

Higher Derivatives

• If $f$ is a differentiable function, then its derivative $f'$ is also a function and may have a derivative of its own, denoted by $f''(x)$.

• $f''(x)$ is the second derivative of $f$.

• Using Leibniz notation, we write the second derivative of $y = f(x)$ as:

  • $\frac{d^2y}{dx^2}$

Example 6: Finding and Interpreting Second Derivatives

• Given $f(x) = x^3 - x$

• First derivative:

• Second derivative:

• $f''(x)$ can be interpreted as the slope of the curve $f'(x)$ at the point $(x, f'(x))$.

• It is the rate of change of the slope of the original curve $y = f(x)$.

Interpretation of Second Derivatives

• In general, a second derivative is a rate of change of a rate of change.

  • If $s = s(t)$ is the position function, $s'(t) = v(t)$ is the velocity function.

  • $v'(t) = a(t)$, acceleration is the second derivative of the position function.

• Acceleration:

  • $a(t) = v'(t) = s''(t)$

  • In Leibniz notation:

Higher Order Derivatives

• The third derivative $f'''(x)$ is the derivative of the second derivative:

• Alternative notations:

• If $s = s(t)$ represents the position, then the third derivative of $s(t)$ is the jerk $j(t)$, which is the rate of change of acceleration.

• Fourth derivative is denoted as , and the n-th derivative is denoted as . If = f(x), we write: