2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple

Welcome back to AP Calculus with Fiveable! We are now diving into some of the most valuable fundamental concepts in calculus that allow us to find derivatives for complex polynomial functions more easily.

🔑 Key Derivative Rules

So far, we’ve only covered the power rule! Be sure to review the power rule before proceeding and learning about the next few derivative rules in this course.

🔄 The Constant Rule of Derivatives

The constant rule states that the derivative of a constant is always zero. Mathematically, if $f(x) = c$, where $c$ is a constant, then $f'(x) = 0$.

For example, the derivative of $f(x) =3$ is 0, or $f'(x)=0$.

➕ The Sum Rule of Derivatives

The sum rule states that the derivative of the sum of two functions is the sum of their derivatives. Mathematically, if $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.

Let’s use $f(x)=2x+3$ as an example. The derivative of $2x$ is $2$ and according to the constant rule, the derivative of $3$ is $0$. Adding these two together, 2+0 is equivalent to 2. Therefore, $f'(x)=2$.

➖ The Difference Rule of Derivatives

The difference rule states that the derivative of the difference of two functions is the difference of their derivatives. Mathematically, if $f(x) = g(x) - h(x)$, then $f'(x) = g'(x) - h'(x)$.

This is very similar to the sum rule, just subtracting rather than adding. Using $f(x)=2x-3$ as an example, we can simply do 2-0 which is 2.

✖️ The Constant Multiple Rule of Derivatives

The constant multiple rule states that the derivative of a constant times a function is the constant multiplied by the derivative of the function. Mathematically, if $f(x) = c \cdot g(x)$, then $f'(x) = c \cdot g'(x)$.

We used this in a previous example by knowing that the derivative of $2x$ is $2(1)=2$.

🏋️‍♂️ Derivative Rules: Practice Problems

Let’s work on a few questions to get the concepts down!

Derivatives: Example 1

Consider the function $f(x) = 2x^2 + 2$. Find the derivative.

Since we see a plus sign, we can quickly identify that we have to use the sum rule to find the derivative of this function. Let’s go through the following steps:

👉 Step 1: Identify the two functions according to the sum rule.

Based on the sum rule $f(x) = g(x) + h(x)$. We can identify that $g(x)$ is $2x^2$ and $h(x)$ is $2$.

👉 Step 2: Take the two derivatives of the two functions, $g(x)$ and $h(x)$, and add them.

$g'(x) =4x$, $h'(x)= 0$. So, $f'(x) =4x+0=4x$.

Great work!

Derivatives: Example 2

Consider the function $f(x) = 100$. Find the derivative.

The number 100 is a constant. So applying the constant rule, which states that the derivative of a constant is zero, $f'(x) = 0.$

Derivatives: Example 3

Consider the function $f(x) = 5(5x+10)$. Find the derivative.

Looking at this function, we can identify that the constant multiple rule should be used.

👉 Step 1: Identify the constant and the function according to the constant multiple rule.

$c=5, g(x) =5x+10$

👉 Step 2: Take the derivative of $g(x)$ and multiply it with the constant.

$g'(x) = 5$, so $f'(x) =5 \cdot 5=25$.

Derivatives: Example 4

Last question! Consider the function $f(x) = 3x^3-6x$. Find the derivative.

👉 Step 1: Identify the two functions according to the difference rule.

Based on the sum rule $f(x) = g(x) - h(x)$, $g(x)$ is $3x^3$ and $h(x)$ is $6x$.

👉 Step 2: Take the two derivatives of the two functions, $g(x)$ and $h(x)$, and subtract them.

$g'(x) =9x^2$, $h'(x)= 6$. So, $f'(x) =9x^2-6$.

🤔 Combining the Power Rule with Other Derivative Rules

Now that you’ve got some practice in with these four new rules, you can combine them with the power rule. On the AP exam, you will likely have to use multiple rules to get the derivative of a given function. Use these two steps to help you with these problems:

1. For each term in the polynomial function, differentiate it using the power rule.
2. Sum or subtract the derivatives of each term according to the original function's operations (addition or subtraction of terms).

Example 1: Finding the Derivative of a Polynomial Function

Consider the following function and solve for $f'(x)$:

By looking at this function, we can see that the sum rule will be used. However, to find the derivative of each individual term, we have to use the power rule and constant rule.

👉 Step 1: Using the Power Rule for each term:

$\frac{d}{dx} (3x^4) = 4 \cdot 3x^{4-1} = 12x^3$

$\frac{d}{dx} (-2x^3) = 3 \cdot (-2)x^{3-1} = -6x^2$

$\frac{d}{dx} (5x^2) = 2 \cdot 5x^{2-1} = 10x$

$\frac{d}{dx} (-7x) = 1 \cdot (-7)x^{1-1} = -7$ 

For the last term, 9, we have to use the constant rule! $\frac{d}{dx} (9) = 0$

👉 Step 2: Combining the Derivatives

Now we can use the sum rule appropriately with each of these terms:

Example 2: Derivative of a Polynomial Function with a Constant Multiple

Consider the following function and solve for $g'(x)$:

👉 Step 1: Using the Power Rule for each term:

$\frac{d}{dx} (2x^5) = 5 \cdot 2x^{5-1} = 10x^4$

$\frac{d}{dx} (-3x^4) = 4 \cdot (-3)x^{4-1} = -12x^3$

$\frac{d}{dx} (6x^3) = 3 \cdot 6x^{3-1} = 18x^2$

👉 Step 2: Combining the Derivatives