Derivative rules
& linearity

Big picture: Additivity and homogeneity give the sum, difference and constant multiple rules. The constant rule is established directly from the derivative definition. Together with the power rule, these results make polynomial differentiation systematic.

RULE 1: THE CONSTANT RULE

The derivative of any constant is zero.

Meaning of constant: $c$ does not change with the differentiation variable $x$. It can be a number such as $\pi$ or a parameter declared constant; a letter is not automatically a variable that changes with $x$.

Why This Makes Sense:

A constant function $f(x)=c$ has a horizontal graph. At an interior input where it is defined nearby, its slope and instantaneous rate are zero.

Examples:

• $\frac{d}{dx}[5]=0$
• $\frac{d}{dx}[-12]=0$
• $\frac{d}{dx}[\pi ]=0$
• $\frac{d}{dx}[0]=0$
• $\frac{d}{dx}[1000000]=0$

Key Takeaway: Constants vanish when you differentiate!

RULE 2: THE CONSTANT MULTIPLE RULE

If $c$ is constant with respect to $x$ and $f$ is differentiable at the input, the derivative of $cf$ is $c{f}^{\prime }$.

Where: $c$ is a constant and $f(x)$ is a differentiable function

In Words: "Pull the constant out front, then differentiate."

Examples:

• $\frac{d}{dx}[7x^3]=7·\frac{d}{dx}[x^3]=7·3x^2=21x^2$
• $\frac{d}{dx}[-4x^5]=-4·5x^4=-20x^4$
• $\frac{d}{dx}[\frac{1}{2}{x}^2]=\frac{1}{2}·2x=x$
• $\frac{d}{dx}[\pi x]=\pi ·1=\pi$

Common Application: When differentiating $a{x}^n$, use constant multiple rule THEN power rule:

RULE 3: THE SUM RULE

If $f$ and $g$ are differentiable at the same input, the derivative of their sum is the sum of their derivatives.

In Words: "Differentiate each term separately, then add the results."

Extends to Multiple Terms:

Examples:

• $\frac{d}{dx}[x^3+x^2]=3x^2+2x$
• $\frac{d}{dx}[x^5+2x]=5x^4+2$
• $\frac{d}{dx}[x^4+x^3+x^2+x]=4x^3+3x^2+2x+1$
• $\frac{d}{dx}[\sqrt{x}+\frac{1}{x}]=\frac{1}{2\sqrt{x}}-\frac{1}{x^2}$

Why it works: The two difference-quotient limits exist as finite numbers, so the limit law permits adding their limits. Do not split a limit into two pieces whose limits fail to exist.

RULE 4: THE DIFFERENCE RULE

If $f$ and $g$ are differentiable at the same input, the derivative of their difference is the difference of their derivatives.

In Words: "Differentiate each term separately, then subtract."

Note: This is really just the Sum Rule with a negative constant multiple:

Examples:

• $\frac{d}{dx}[x^3-x^2]=3x^2-2x$
• $\frac{d}{dx}[5x^4-2x]=20x^3-2$
• $\frac{d}{dx}[x^2-7]=2x-0=2x$ (constant disappears!)
• $\frac{d}{dx}[\frac{1}{x}-\sqrt{x}]=-\frac{1}{x^2}-\frac{1}{2\sqrt{x}}$

Watch Signs: Keep track of negative signs carefully!

Domains in the root and reciprocal examples: Both $\sqrt{x}+\frac{1}{x}$ and $\frac{1}{x}-\sqrt{x}$, and their displayed derivative formulas, require $x>0$. The reciprocal excludes zero; the square root excludes negative inputs. The polynomial examples have all real inputs as their domains.

Apply the power rule on its valid domain: For $a{x}^n$, the constants $a$ and $n$ do not change with $x$. For arbitrary real exponents the formula applies on positive inputs; extend it to other inputs only with appropriate real-power and differentiability checks.

The Power of Linearity: Using All Rules Together

For ANY polynomial (or linear combination of differentiable functions), you can:

1. Break it into terms using Sum/Difference Rules
2. Pull out constants using Constant Multiple Rule
3. Apply Power Rule (or other derivative formulas) to each term
4. Constants disappear using Constant Rule
5. Combine results using Sum/Difference Rules again

General Formula for Polynomials:

Difference rule: Replace $g$ by $-g$ in the sum rule and apply the constant multiple rule with scalar $-1$. The constant rule has its own direct proof below; an expression involving $x^{-1}$ would not justify a derivative at zero.

Hypotheses behind the proofs: Work at a common interior input where the component functions are defined nearby and differentiable. Their difference-quotient limits are finite. This is what permits separating a sum limit or pulling out a fixed scalar. The statements give a sufficient rule; they are not a test that each component must be differentiable whenever the sum is.

Workflow for a finite sum of powers: Identify the variable and domain first. More general products, quotients and compositions can require other rules.

1. REWRITE: Convert all roots and fractions to exponent form
2. EXPAND: Multiply out any products (if simple enough)
3. COMBINE: Add/subtract like terms to simplify
4. SEPARATE: Identify each term (use Sum/Difference Rules)
5. DIFFERENTIATE: Apply Constant Multiple Rule + Power Rule to each term
6. SIMPLIFY: Combine results and clean up notation
7. VERIFY: For a nonconstant polynomial, check that the degree drops by one. For roots and reciprocals, check coefficient arithmetic and the differentiability domain instead.

Acronym for the Four Rules:

"CSCD" - Constant, Sum, Constant Multiple, Difference

Rhyme to Remember:

"Constants are zero, pull numbers out,
Add derivatives when adding throughout.
Subtract derivatives when you subtract—
Linearity makes differentiation exact!"

• Mistake 1: Confusing a constant with a constant times the independent variable. The correct derivatives are $\frac{d}{dx}[5x]=5$ and $\frac{d}{dx}[5]=0$.
• Mistake 2: Losing the minus sign belonging to a term. For $x^2-3x$, keep the first derivative positive and subtract the derivative of the second term: $\frac{d}{dx}[x^2-3x]=2x-3$. Do not subtract both derivatives.
• Mistake 3: Treating a product as a sum. Here $(x^2)(x^3)=x^5$ by exponent laws, so one power rule gives $5x^4$. The product rule is another valid method; multiplying the individual derivatives would be incorrect.
• Mistake 4: Losing constant multiples: $\frac{d}{dx}[6x^3]=18x^2$, not $3x^2$
• Mistake 5: Leaving avoidable clutter. For example, $2x+3x=5x$. Both forms are equivalent, but the combined expression is clearer; follow any requested final form.
• Mistake 6: Assuming the product of derivatives is a general product formula. In general, $\frac{d}{dx}[fg]=f^{\prime }g+f{g}^{\prime }$, not merely $f^{\prime }{g}^{\prime }$. For example, differentiating $x·x=x^2$ gives $2x$, whereas multiplying the individual derivatives gives one.
• Mistake 7: Forgetting that $x$ has an invisible coefficient of 1
• Mistake 8: Ignoring a composite inner function. The chain rule is introduced in Unit 3. For a simple polynomial composition you may expand first; otherwise an inner derivative is needed.

How to write a clear AP-style solution:

• Show rule application: On FRQ, briefly indicate which rules you're using
• Use a clear final form: Combine like terms and simplify coefficients when useful or requested. Equivalent exact expressions can represent the same derivative.
• Work systematically: Go term-by-term to avoid missing anything
• Check signs carefully: A minus sign stays attached to its term, but it can multiply a negative exponent coefficient. For example, $\frac{d}{dx}[-2x^{-2}]=4x^{-3}$.
• Show the requested reasoning: Give enough differentiation steps to support the result. If the task asks for a proof or the limit definition, an unsupported shortcut does not establish it.
• Recognize the structure: Linearity applies to sums, differences and constant multiples. A product, quotient or composition of variable functions requires valid algebra or an additional differentiation rule.
• Equivalent representations: $\frac{3}{2}x=1.5x$. Exact fraction notation avoids unintended rounding; use any precision or form specified in the task.

Common FRQ Formats:

1. Find $f^{\prime }(x)$ for a specified function.
2. Show a claimed derivative identity for $f(x)$, providing the calculation or proof requested.
3. Find a tangent slope by evaluating $f^{\prime }(a)$ at an allowed input $a$.
4. "Determine the rate of change..." (find and evaluate derivative)
5. Find inputs satisfying a stated derivative equation: differentiate, solve the equation, then check the original domain.
6. Find a tangent equation using both the point $(a,f(a))$ and the slope $f^{\prime }(a)$.

Topic 2.6 connects to:

• Topic 2.5: The power rule differentiates each power; linearity combines those derivatives with their original constant coefficients.
• Topics 2.7–2.10: Derivatives of trigonometric, exponential and logarithmic functions can be combined linearly. Product and quotient rules handle other structures; the chain rule comes in Unit 3.
• Unit 3: Composite, implicit and inverse differentiation use these rules within larger calculations.
• Unit 4: Motion problems use derivatives of position functions
• Units 5 and 6: Unit 5 applies derivatives to critical points, graph behavior and optimization. Unit 6 develops integration and accumulation, with its own linearity properties and links to differentiation.
• Beyond polynomials: A linear combination can include any differentiable component functions; the component derivative formulas may come from other topics.

Remember: The constant rule gives $\frac{d}{dx}[c]=0$. The other three rules are captured by $(af+bg{)}^{\prime }=a{f}^{\prime }+b{g}^{\prime }$ when the coefficients are constant and both functions are differentiable at the input. Differentiate each component, retain its coefficient and sign, and combine the results. Check the domain and function structure before applying the rule.