Defining the Derivative of a Function and Using Derivative Notation - AP Calculus AB Study Guide

The Core Idea: Defining the Derivative of a Function and Using Derivative Notation

The central concept of this topic is the formal definition of the derivative. While we may intuitively understand the derivative as the slope of a tangent line or an instantaneous rate of change, this topic provides the rigorous mathematical foundation for that idea. The derivative is not simply a rule to be memorized; it is fundamentally a limit.

The process involves starting with the average rate of change between two points on a curve, represented by the slope of a secant line. This average rate is expressed as a difference quotient. By taking the limit of this difference quotient as the distance between the two points approaches zero, we transition from an average rate of change over an interval to an instantaneous rate of change at a single point. This limit, if it exists, is the derivative of the function.

Key Formulas & Definitions

The derivative of a function $f$ is defined as the limit of a difference quotient. There are two primary forms for this definition.

The Derivative as a General Function

This form is used to find the derivative function, $f^{\prime }(x)$, which gives the slope of the tangent line at any point $x$ in the domain of $f^{\prime }$.

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

The Derivative at a Specific Point

This alternate form is often used to find the derivative at a single specific point, $x=a$.

$$f^{\prime }(a)=\underset{x\to a}{\lim }\frac{f(x)-f(a)}{x-a}$$

Derivative Notation

The derivative of a function $y=f(x)$ can be expressed using several different notations. The most common notations in AP Calculus are:

• Lagrange's Notation:$f^{\prime }(x)$ (read "f prime of x")

• Leibniz's Notation:$\frac{dy}{dx}$ (read "dy dx")

• Euler's Notation:$y^{\prime }$ (read "y prime")

Understanding the Limit Definition

The two forms of the limit definition of the derivative are derived from the same geometric concept: the slope of a secant line approaching the slope of a tangent line.

• The $h\to 0$ Definition: The expression $\frac{f(x+h)-f(x)}{h}$ represents the slope of the secant line between the points $(x,f(x))$ and $(x+h,f(x+h))$. Here, $h$ is the small horizontal distance between the two points. The limit ${\lim }_{h\to 0}$ signifies that we are sliding the second point along the curve closer and closer to the first point. As the distance $h$ shrinks to zero, the slope of the secant line becomes the slope of the tangent line at the point $(x,f(x))$.

• The $x\to a$ Definition: The expression $\frac{f(x)-f(a)}{x-a}$ is the standard slope formula between a fixed point $(a,f(a))$ and a variable point $(x,f(x))$. The limit ${\lim }_{x\to a}$ signifies that we are moving the variable point $(x,f(x))$ along the curve toward the fixed point $(a,f(a))$. As $x$ gets infinitely close to $a$, this slope also approaches the slope of the tangent line at $x=a$.

Core Concepts & Rules

• The derivative of a function $f$ is defined as the limit of its difference quotient.

• The difference quotient can be written as $\frac{f(x+h)-f(x)}{h}$ or $\frac{f(x)-f(a)}{x-a}$.

• The derivative $f^{\prime }(x)$ is a function that gives the instantaneous rate of change of $f(x)$ with respect to $x$.

• The value of the derivative at a point, $f^{\prime }(a)$, represents the slope of the line tangent to the graph of $f(x)$ at $x=a$.

• The derivative of $y=f(x)$ can be denoted as $f^{\prime }(x)$, $y^{\prime }$, or $\frac{dy}{dx}$.

• Finding the derivative of a function using its definition is an algebraic process that requires simplifying the difference quotient until the limit can be evaluated directly.

Step-by-Step Example 1: Finding the Derivative Function

Problem: Use the limit definition of the derivative to find $f^{\prime }(x)$ for the function $f(x)=2x^2-5x$.

Step 1: Set up the difference quotient $\frac{f(x+h)-f(x)}{h}$.

First, find the expression for $f(x+h)$ by substituting $(x+h)$ for every $x$ in the original function.

$f(x+h)=2(x+h{)}^2-5(x+h)$

Now, substitute $f(x+h)$ and $f(x)$ into the difference quotient formula.

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{[2(x+h{)}^2-5(x+h)]-[2x^2-5x]}{h}$$

Step 2: Expand and simplify the numerator.

Be careful with algebraic expansion and distribution of the negative sign.

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{[2(x^2+2xh+h^2)-5x-5h]-2x^2+5x}{h}$$

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{2x^2+4xh+2h^2-5x-5h-2x^2+5x}{h}$$

Combine like terms in the numerator. Notice that all terms without an $h$ should cancel out.

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{4xh+2h^2-5h}{h}$$

Step 3: Factor out $h$ from the numerator and cancel.

The goal is to eliminate the $h$ in the denominator, which causes the $\frac{0}{0}$ indeterminate form.

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h(4x+2h-5)}{h}$$

$$f^{\prime }(x)=\underset{h\to 0}{\lim }(4x+2h-5)$$

Step 4: Evaluate the limit by substituting $h=0$.

Now that the expression is no longer an indeterminate form, we can use direct substitution.

$$f^{\prime }(x)=4x+2(0)-5$$

$$f^{\prime }(x)=4x-5$$

Answer: The derivative of $f(x)=2x^2-5x$ is $f^{\prime }(x)=4x-5$.

Step-by-Step Example 2: Finding the Derivative at a Point

Problem: Use the alternate limit definition of the derivative to find the slope of the tangent line to the graph of $g(x)=\sqrt{x+1}$ at $x=3$.

Step 1: Set up the alternate difference quotient $\frac{g(x)-g(a)}{x-a}$.

Here, our function is $g(x)=\sqrt{x+1}$ and our point is $a=3$. First, find $g(a)=g(3)$.

$g(3)=\sqrt{3+1}=\sqrt{4}=2$

Now, set up the limit.

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{g(x)-g(3)}{x-3}=\underset{x\to 3}{\lim }\frac{\sqrt{x+1}-2}{x-3}$$

Step 2: Use an algebraic technique to simplify the expression.

Direct substitution of $x=3$ yields $\frac{\sqrt{4}-2}{3-3}=\frac{0}{0}$, an indeterminate form. For expressions with square roots, the standard technique is to multiply the numerator and denominator by the conjugate of the numerator. The conjugate of $\sqrt{x+1}-2$ is $\sqrt{x+1}+2$.

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{\sqrt{x+1}-2}{x-3}\cdot \frac{\sqrt{x+1}+2}{\sqrt{x+1}+2}$$

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{(\sqrt{x+1}{)}^2-2^2}{(x-3)(\sqrt{x+1}+2)}$$

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{(x+1)-4}{(x-3)(\sqrt{x+1}+2)}$$

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{x-3}{(x-3)(\sqrt{x+1}+2)}$$

Step 3: Cancel the common factor and evaluate the limit.

The $(x-3)$ term cancels, removing the indeterminate form.

$$g^{\prime }(3)=\underset{x\to 3}{\lim }\frac{1}{\sqrt{x+1}+2}$$

Now, use direct substitution for $x=3$.

$$g^{\prime }(3)=\frac{1}{\sqrt{3+1}+2}=\frac{1}{\sqrt{4}+2}=\frac{1}{2+2}=\frac{1}{4}$$

Answer: The slope of the tangent line to $g(x)=\sqrt{x+1}$ at $x=3$ is $\frac{1}{4}$.

Using Your Calculator

This topic is purely analytical, meaning problems will require you to show the algebraic steps of using the limit definition. A calculator cannot perform these symbolic manipulations for you.

However, a graphing calculator can be an excellent tool for checking your answer. After you have found the derivative at a point by hand, you can use the calculator's numerical derivative feature to approximate the value and verify your result.

Example: To check the answer from Example 2 ($g^{\prime }(3)=\frac{1}{4}$ for $g(x)=\sqrt{x+1}$):

1. On the home screen of a TI-84 style calculator, press MATH and select `nDeriv((or $8).

2. The syntax is nDeriv(function, variable, value).

3. Enter the following: nDeriv(√(X+1), X, 3)

4. The calculator will return $0.25$, which confirms that the answer of $\frac{1}{4}$ is correct.

AP Exam Quick Hit

Common Question Types

• Direct Calculation: You will be asked to use the limit definition of the derivative to find $f^{\prime }(x)$ for a polynomial or simple rational function. This is a common free-response question part.

  • Example: "Let $f(x)=\frac{1}{x+2}$. Find $f^{\prime }(x)$ using the limit definition of the derivative."

• Recognizing the Definition: A multiple-choice question will present a limit and ask you to identify it as the derivative of a specific function at a specific point.

  • Example: "The limit ${\lim }_{h\to 0}\frac{\cos (\pi +h)-\cos (\pi )}{h}$ represents $f^{\prime }(a)$ for what function $f$ and what value $a$?" (Answer: $f(x)=\cos (x)$ and $a=\pi$)

• Alternate Form Calculation: You may be asked to find the derivative at a point for a function involving a square root, which requires using the alternate form and multiplying by the conjugate.

  • Example: "Find the slope of the line tangent to $f(x)=\sqrt{2x}$ at $x=8$ using the limit definition."

Common Mistakes

• Algebraic Errors: The most common mistakes are purely algebraic. This includes incorrectly expanding $(x+h{)}^n$, making sign errors when distributing a negative, or incorrectly combining terms.

• Forgetting Limit Notation: Dropping the ${\lim }_{h\to 0}$ or ${\lim }_{x\to a}$ notation in the intermediate steps of your work. The limit notation is required on every line until the limit is actually evaluated.

• Premature Substitution: Plugging in $h=0$ or $x=a$ before algebraically canceling the term in the denominator that is causing the $\frac{0}{0}$ indeterminate form.

• Conceptual Errors with $f(x+h)$: Incorrectly forming the $f(x+h)$ term. For example, for $f(x)=x^2$, writing $x^2+h$ instead of the correct $(x+h{)}^2$.

• Confusing the Two Definitions: Using the $h\to 0$ form when the $x\to a$ form would be more efficient (or vice-versa), or mixing components of the two different formulas.