Lecture #7: The Derivative Function & Interpretations

MATH 145 – Calculus I

Torin Quinlivan
tquinlivan@knox.edu

Knox College

September 29, 2025

Example: The Derivative at a Point

Example: Find \(f'(2)\) for \(f(x) = x^2\).

The Derivative Function

What if we do this for all \(x\) at once, symbolically, instead of just \(x = 2\)?

This leads us to the following definition:

Let \(f\) be a function and \(x\) a value in the function’s domain. We define the derivative of \(f\), a new function called \(f'\), by the formula \[ f'(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}, \]

provided this limit exists.

\(f'(x)\) is sometimes written as \(\dfrac{d}{dx} f(x)\).

Example: The Derivative Function

Examples:

  1. Find \(f'(x)\) for \(f(x) = x^2\). Use this function formula to find \(f'(2)\).
  2. Find \(g'(x)\) for \(g(x) = 6\).
  3. Find \(h'(x)\) for \(h(x) = 4x - 2\).
  4. Find \(\ell'(x)\) for \(\ell(x) = ax^2 + bx + c\).

Example: The Derivative Function

Example: Let \(f(x) = 3x^2 + 6\).

  1. Use the derivative at a point formula to find \(f'(-2)\).
  2. Use the derivative at a point formula to find \(f'(6)\).
  3. Use the definition of the derivative formula to find \(f'(x)\).
  4. Find \(f'(-2)\) with the derivative function.
  5. Find \(f'(6)\) with the derivative function.

Example: The Derivative Function

Example: Let \(g(x) = \dfrac{1}{x}\).

  1. Use the derivative at a point formula to find \(g'(-3)\).
  2. Use the derivative at a point formula to find \(g'(1)\).
  3. Use the definition of the derivative formula to find \(g'(x)\).
  4. Find \(g'(-3)\) with the derivative function.
  5. Find \(g'(1)\) with the derivative function.

The Derivative Function, Graphically

Example: Sketch the graph of \(q'(x)\) if \(q(x)\) has the graph below.

Application of the Derivative

Recall:

We say that \(f\) is increasing on \((a, b)\) provided that the function is always rising as we move from left to right. That is, for any \(x\) and \(y\) in \((a,b)\), if \(x < y\), then \(f(x) < f(y)\).

We say that \(f\) is decreasing on \((a, b)\) provided that the function is always falling as we move from left to right. That is, for any \(x\) and \(y\) in \((a,b)\), if \(x < y\), then \(f(x) > f(y)\).

Application of the Derivative

It can then be shown that:

Let \(f\) be a function that is differentiable on an interval \((a, b)\). If \(f'(x) > 0\) for every \(x\) such that \(a < x < b\), then \(f\) is increasing on \((a, b)\).

Let \(f\) be a function that is differentiable on an interval \((a, b)\). If \(f'(x) < 0\) for every \(x\) such that \(a < x < b\), then \(f\) is decreasing on \((a, b)\).

The Derivative Function, Graphically

Example: Given the graph of \(r'(x)\) below, give a rough sketch of \(r(x)\).