Derivative Rules: Constant, Sum, Difference, and Constant Multiple - AP Calculus AB Study Guide

The Core Idea: Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Finding the derivative of a function using the limit definition can be a lengthy and algebraically intensive process. The fundamental purpose of derivative rules is to provide a more efficient and systematic method for calculating derivatives. This topic introduces the first set of these foundational rules, which allow us to differentiate functions that are constructed by adding, subtracting, or scaling simpler functions.

By understanding how to differentiate constants, sums, differences, and constant multiples of functions, we can move from the conceptual definition of the derivative to a practical, operational toolkit. These rules form the building blocks for differentiating more complex functions, such as polynomials, and are essential for nearly all derivative calculations that follow. They allow us to break down a complicated function into manageable parts, differentiate each part individually, and then combine the results to find the derivative of the whole.

Key Rules

The following rules are used to find the derivatives of functions formed from the combination or scaling of other differentiable functions. Let $c$ be a constant, and let $f(x)$ and $g(x)$ be differentiable functions.

1. The Constant Rule

The derivative of any constant function is zero. This is because a constant function represents a horizontal line, and the slope of a horizontal line is always zero.

$$\frac{d}{dx}(c)=0$$

2. The Constant Multiple Rule

The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This rule allows us to "factor out" a constant coefficient before differentiating.

$$\frac{d}{dx}[c\cdot f(x)]=c\cdot \frac{d}{dx}[f(x)]=c\cdot f^{\prime }(x)$$

3. The Sum Rule

The derivative of a sum of two functions is the sum of their individual derivatives. This rule allows us to differentiate a function term-by-term.

$$\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}[f(x)]+\frac{d}{dx}[g(x)]=f^{\prime }(x)+g^{\prime }(x)$$

4. The Difference Rule

The derivative of a difference of two functions is the difference of their individual derivatives. Similar to the Sum Rule, this allows for term-by-term differentiation.

$$\frac{d}{dx}[f(x)-g(x)]=\frac{d}{dx}[f(x)]-\frac{d}{dx}[g(x)]=f^{\prime }(x)-g^{\prime }(x)$$

Understanding the Combination of Rules

The true power of these rules is realized when they are used in combination, particularly when differentiating polynomial functions. A function like $h(x)=5x^3-2x+9$ is a combination of sums, differences, and constant multiples of simpler functions (namely $x^3$, $x$, and $9$).

To find the derivative $h^{\prime }(x)$, we do not need to resort to the limit definition. Instead, we can apply the rules systematically:

1. First, use the Sum and Difference Rules to break the function into individual terms:

   $$\frac{d}{dx}(5x^3-2x+9)=\frac{d}{dx}(5x^3)-\frac{d}{dx}(2x)+\frac{d}{dx}(9)$$

2. Next, apply the Constant Multiple Rule to the terms with coefficients:

   $$=5\cdot \frac{d}{dx}(x^3)-2\cdot \frac{d}{dx}(x)+\frac{d}{dx}(9)$$

3. Finally, apply the Power Rule (from Topic 2.5) and the Constant Rule to each term:

   $$=5\cdot (3x^2)-2\cdot (1)+0$$

4. Simplify the result:

   $$h^{\prime }(x)=15x^2-2$$

This process demonstrates that these four rules provide a complete algorithm for differentiating any polynomial function by handling each term individually.

Core Concepts & Rules

• Derivative of a Constant is Zero: The rate of change of a constant quantity is always zero. For any constant $c$, $\frac{d}{dx}(c)=0$.

• Constants Factor Out: When differentiating, constant coefficients can be factored out of the derivative operation. If $c$ is a constant, $\frac{d}{dx}[c\cdot f(x)]=c\cdot f^{\prime }(x)$.

• Derivatives Distribute Over Addition: The derivative of a sum is the sum of the derivatives. You can differentiate a function term-by-term if the terms are added together. $\frac{d}{dx}[f(x)+g(x)]=f^{\prime }(x)+g^{\prime }(x)$.

• Derivatives Distribute Over Subtraction: The derivative of a difference is the difference of the derivatives. You can also differentiate term-by-term if the terms are subtracted. $\frac{d}{dx}[f(x)-g(x)]=f^{\prime }(x)-g^{\prime }(x)$.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $f(x)=-2x^4+8x^3-11$.

Solution:

Step 1: Apply the Sum and Difference Rules

Break the function into three separate terms to be differentiated.

$$f^{\prime }(x)=\frac{d}{dx}(-2x^4+8x^3-11)=\frac{d}{dx}(-2x^4)+\frac{d}{dx}(8x^3)-\frac{d}{dx}(11)$$

Step 2: Apply the Constant Multiple Rule

For the first two terms, factor out the constant coefficients. The third term is a constant, so we will apply the Constant Rule to it in the next step.

$$f^{\prime }(x)=-2\cdot \frac{d}{dx}(x^4)+8\cdot \frac{d}{dx}(x^3)-\frac{d}{dx}(11)$$

Step 3: Differentiate Each Term

Use the Power Rule ($\frac{d}{dx}{x}^n=n{x}^{n-1}$) for the $x^4$ and $x^3$ terms, and the Constant Rule for the $11$.

$$f^{\prime }(x)=-2\cdot (4x^{4-1})+8\cdot (3x^{3-1})-0$$

$$f^{\prime }(x)=-2(4x^3)+8(3x^2)-0$$

Step 4: Simplify the Expression

Perform the multiplication to get the final derivative function.

$$f^{\prime }(x)=-8x^3+24x^2$$

Step-by-Step Example 2: Exam-Style Application

Problem: The functions $f$ and $g$ are differentiable. The table below gives values of $f(x)$, $g(x)$, $f^{\prime }(x)$, and $g^{\prime }(x)$ for selected values of $x$.

If $h(x)=4f(x)-3g(x)+7$, what is the value of $h^{\prime }(1)$?

Solution:

Step 1: Find the derivative function $h^{\prime }(x)$ using the derivative rules.

First, apply the Sum and Difference Rules to differentiate $h(x)$ term-by-term.

$$h^{\prime }(x)=\frac{d}{dx}[4f(x)-3g(x)+7]=\frac{d}{dx}[4f(x)]-\frac{d}{dx}[3g(x)]+\frac{d}{dx}[7]$$

Step 2: Apply the Constant Multiple and Constant Rules.

Factor out the constants $4$ and $3$, and take the derivative of the constant $7$.

$$h^{\prime }(x)=4\cdot \frac{d}{dx}[f(x)]-3\cdot \frac{d}{dx}[g(x)]+0$$

$$h^{\prime }(x)=4f^{\prime }(x)-3g^{\prime }(x)$$

Step 3: Evaluate $h^{\prime }(x)$ at the specified point, $x=1$.

Substitute $x=1$ into the expression for $h^{\prime }(x)$.

$$h^{\prime }(1)=4f^{\prime }(1)-3g^{\prime }(1)$$

Step 4: Use the values from the table to find the final answer.

From the table, we find that $f^{\prime }(1)=5$ and $g^{\prime }(1)=-3$. Substitute these values into the equation.

$$h^{\prime }(1)=4(5)-3(-3)$$

$$h^{\prime }(1)=20-(-9)$$

$$h^{\prime }(1)=20+9=29$$

Using Your Calculator

The rules in this topic are purely analytical, meaning they are performed by hand to find a derivative function. A calculator cannot derive a function symbolically (e.g., it cannot turn $x^2$ into $2x$).

However, a graphing calculator is an excellent tool for checking your answer at a specific point. After you have analytically found the derivative function, you can use the calculator's numerical derivative feature to verify its value.

Example: To check the result from Example 1, $f^{\prime }(x)=-8x^3+24x^2$, at $x=2$.

1. By Hand: Calculate f'(2) = -8(2)^3 + 24(2)^2 = -8(8) + 24(4) = -64 + 96 = 32`. 2. **On Calculator (TI-84 Style):** * Press `MATH` and select `8: nDeriv(`. * Enter the expression: $nDeriv(Y₁, X, 2) where $Y_1$ contains the original function $f(x)=-2x^4+8x^3-11$.

   • Alternatively, type the function directly: nDeriv(-2X^4 + 8X^3 - 11, X, 2).

   • The calculator will return $32$, confirming that your analytical derivative is likely correct.

AP Exam Quick Hit

Common Question Types

• Direct Differentiation of Polynomials: You will be asked to find the derivative of a polynomial function.

  • Example: Find $\frac{dy}{dx}$ if $y=\frac{1}{2}{x}^4-3x^2+\pi$.

• Table-Based Problems: You will be given a table of function and derivative values and asked to find the derivative of a combination of those functions at a specific point.

  • Example: Given the table in Example 2 above, find the derivative of $k(x)=2g(x)-f(x)$ at $x=3$.

• Finding the Slope of a Tangent Line: You will be asked to find the slope of the tangent line to a function at a point, which requires finding the value of the derivative at that point.

  • Example: What is the slope of the line tangent to the graph of $f(x)=2x^3-x+5$ at $x=-1$?

Common Mistakes

• Mistaking the Constant Rule for the Constant Multiple Rule: A common error is to set the derivative of a term like $5x^3$ to zero, confusing the Constant Multiple Rule with the Constant Rule. The derivative is $5\cdot (3x^2)=15x^2$, not $0$.

• Forgetting the Derivative of a Constant is Zero: When differentiating a polynomial, students sometimes carry the constant term over to the derivative. For $f(x)=x^2+6$, the derivative is $f^{\prime }(x)=2x$, not $2x+6$.

• Applying Sum/Difference Rules to Products or Quotients: These rules only apply to sums and differences. Students may incorrectly assume $\frac{d}{dx}(f(x)\cdot g(x))=f^{\prime }(x)\cdot g^{\prime }(x)$. This is incorrect; separate rules (the Product and Quotient Rules) are required for these cases.

• Algebraic Errors: After correctly applying the calculus rules, simple arithmetic mistakes during simplification (e.g., $4\cdot 3=7$ or errors with negative signs) can lead to an incorrect final answer. Always double-check your algebra.