Why These Rules Matter
Once you understand what a derivative means, the next step is learning rules that make differentiation fast. The constant rule, sum rule, and difference rule are among the simplest rules in AP Calculus, but they are also some of the most frequently used.
The key idea is that you do not have to differentiate an entire expression all at once. If a function is made from several terms added or subtracted together, you can differentiate each term separately and then combine the results.
For example, consider . Instead of treating this as one large expression, differentiate each term independently.
The Constant Rule
A constant is a number that does not change as changes. Its graph is a horizontal line, so its slope is always zero.
In other words, the derivative of any constant is zero. This includes positive numbers, negative numbers, fractions, and decimals.
Differentiate .
Why is the derivative zero?
Think about the graph of . It is a horizontal line. A horizontal line has slope zero everywhere, so its derivative must be zero.
The Sum Rule
The sum rule says that the derivative of a sum is the sum of the derivatives. You can differentiate each term separately and then add the results.
This is especially useful for polynomial functions, because a polynomial is simply a collection of terms added together.
Differentiate .
The sum rule works for any functions
The rule is not limited to polynomials. If and are differentiable functions, then their sum can be differentiated term-by-term.
Differentiate .
The Difference Rule
The difference rule is essentially the same idea as the sum rule. When functions are subtracted, differentiate each term separately and keep the subtraction.
The important point is that the negative sign stays attached to the term being subtracted.
Differentiate .
Combining All Three Rules
In practice, these rules usually appear together. A function may contain positive terms, negative terms, and constants all in the same expression. The strategy is simple: differentiate one term at a time.
Differentiate .
Constants Can Be Coefficients
A common point of confusion is the difference between a constant term and a constant coefficient. In , the number 7 is a constant coefficient multiplying ; the entire term is not a constant.
You therefore do not differentiate the 7 by itself and stop. Instead, the constant multiple stays in front while you differentiate the function it multiplies.
Differentiate .
A Useful Shortcut for Polynomials
For a polynomial, you can combine the power rule with the sum and difference rules. Differentiate every term independently, reduce each exponent by 1, multiply by the original exponent, and eliminate constant terms.
Differentiate .
Common Mistakes
A standalone constant such as has derivative zero. But a coefficient such as the 5 in stays with the term while you differentiate .
The difference rule preserves subtraction. For example, when differentiating , the derivative of the second term is still subtracted: .
The sum and difference rules tell you to differentiate every term. Do not stop after finding the derivative of the first term in a polynomial.
The derivative of is zero, but the derivative of is not zero. The presence of means the term changes as changes.
Practice Problems
Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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Differentiate .
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