AP Calculus AB

## Learning Outcomes

- Differentiate polynomial functions using the power rule.
- Apply constant multiple, sum, and difference rules to polynomial expressions.
- Identify the derivative of constant, linear, and higher-degree polynomial terms.
- Find slopes of tangent lines to polynomial graphs at specified points.
- Use polynomial derivatives to locate horizontal tangents and critical numbers.
- Avoid frequent algebra errors involving coefficients, signs, and rewritten polynomial forms.

## AP Calculus AB Syllabus

In AP Calculus AB, polynomial differentiation belongs to the derivatives content area. The derivative is developed from limits and then used as an instantaneous rate of change. For this subtopic, you should be able to move fluently from a polynomial formula to its derivative and then interpret that derivative in slope, tangent-line, or rate-of-change questions.

- Know that a derivative gives instantaneous rate of change.
- Use derivative notation such as $$f'(x)$$, $$\frac{dy}{dx}$$, and $$\frac{d}{dx}[f(x)]$$.
- Apply the power rule to terms of the form $$ax^n$$ where $$n$$ is a nonnegative integer in polynomial work.
- Use the constant multiple rule and sum/difference rules to differentiate term by term.
- Recognize that the derivative of a constant is $$0$$.
- Evaluate derivatives at specific $$x$$-values to find slopes.
- Write equations of tangent lines to polynomial graphs.
- Use derivatives of polynomial functions in no-calculator and calculator-permitted AP-style problems.

## Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, look more closely at that area during your revision.

1. Find $$\frac{d}{dx}(5x^4-3x^2+7x-11)$$.
2. If $$f(x)=2x^3-9x^2+12x$$, for what values of $$x$$ does the graph of $$f$$ have a horizontal tangent?
3. Find the equation of the tangent line to $$y=x^3-4x+1$$ at $$x=2$$.
4. Explain why the derivative of $$8$$ is $$0$$ but the derivative of $$8x$$ is $$8$$.
5. Differentiate $$g(x)=3(x-1)(x+2)$$ without using the product rule.

## Introduction

Polynomials are among the most friendly functions in AP Calculus AB. They are continuous for all real numbers, differentiable for all real numbers, and their derivatives can be found quickly using the power rule and basic derivative properties. Because many AP questions mix concepts, a simple polynomial derivative may appear inside a tangent-line problem, a motion problem, a graph interpretation problem, or a free-response justification.

The main skill is term-by-term differentiation. Once a polynomial is written as a sum or difference of powers of $$x$$, each term can be differentiated separately. This makes the power rule one of the fastest and most useful derivative rules in early calculus.

> **Key Term: Derivative**
> The derivative of a function gives its instantaneous rate of change. At a point on a graph, it gives the slope of the tangent line, when that tangent slope exists.

### From the derivative definition to the power rule

The derivative begins with a limit of a difference quotient. For a function $$f$$, the derivative at $$x$$ is defined by

f'(x)=\lim\_{h\to 0}\frac{f(x+h)-f(x)}{h}
when this limit exists. The power rule is a shortcut that comes from this definition. For a positive integer $$n$$,

$$
\frac{d}{dx}(x^n)=nx^{n-1}
$$

For example,

$$
\frac{d}{dx}(x^5)=5x^4
$$

The exponent becomes a coefficient, and the exponent decreases by $$1$$.

> **Key Term: Power rule**
> The derivative rule $$\frac{d}{dx}(x^n)=nx^{n-1}$$. For polynomial terms, $$n$$ is a nonnegative integer.

A quick reason this rule works can be seen from the derivative definition. Suppose $$f(x)=x^n$$. Then

f'(x)=\lim\_{h\to 0}\frac{(x+h)^n-x^n}{h}
When $$(x+h)^n$$ is expanded, the first two terms are

$$
(x+h)^n=x^n+nx^{n-1}h+\text{terms containing higher powers of }h
$$

Subtracting $$x^n$$ cancels the first term. Dividing by $$h$$ leaves

$$
nx^{n-1}+\text{terms containing }h
$$

As $$h\to 0$$, the terms still containing $$h$$ go to $$0$$, leaving $$nx^{n-1}$$. On the AP exam, you usually use the rule directly rather than proving it.

> **Test Tip:**
> If a question asks you to find a derivative of a polynomial, do not return to the limit definition unless the question specifically asks for a limit-based derivation. The power rule is faster and less error-prone.

The rule also handles linear and constant-looking powers:

$$
\frac{d}{dx}(x)=\frac{d}{dx}(x^1)=1x^0=1
$$

Since $$x^0=1$$, this gives the familiar result that the derivative of $$x$$ is $$1$$.

For constants, think of $$7$$ as a number that does not change as $$x$$ changes. Its derivative is

$$
\frac{d}{dx}(7)=0
$$

If you use exponent language, a nonzero constant such as $$7$$ can be written as $$7x^0$$, but applying the power rule directly to $$x^0$$ gives

$$
\frac{d}{dx}(7x^0)=7\cdot 0x^{-1}=0
$$

The practical result is simple: constant terms disappear when you differentiate.

### Differentiating polynomial terms and full polynomial functions

A polynomial is a finite sum of terms involving nonnegative integer powers of $$x$$. Examples include $$4x^3-2x+9$$, $$x^5+7x^2$$, and $$-6$$.

> **Key Term: Polynomial**
> A function that can be written as a finite sum of terms $$a_nx^n$$, where each exponent $$n$$ is a nonnegative integer and each coefficient $$a_n$$ is a real number.

Polynomial differentiation depends on three basic facts:

$$
\frac{d}{dx}(x^n)=nx^{n-1}
$$

\frac{d}{dx}(c f(x))=c f'(x)
\frac{d}{dx}(f(x)\pm g(x))=f'(x)\pm g'(x)

> **Key Term: Constant multiple rule**
> A derivative rule stating that a constant factor can be kept in front of the derivative: $$\frac{d}{dx}[c f(x)]=c f'(x)$$.
>
> **Key Term: Sum and difference rules**
> Derivative rules stating that the derivative of a sum or difference is the sum or difference of the derivatives.

These rules allow you to differentiate a polynomial one term at a time.

> **Question:**
> Differentiate $$f(x)=4x^5-3x^2+7x-9$$.
> Solution:
>
> $$
> f'(x)=\frac{d}{dx}(4x^5)-\frac{d}{dx}(3x^2)+\frac{d}{dx}(7x)-\frac{d}{dx}(9)
> $$
>
> Apply the power rule to each variable term:
>
> $$
> f'(x)=4(5x^4)-3(2x)+7(1)-0
> $$
>
> Therefore,
>
> $$
> f'(x)=20x^4-6x+7
> $$

Notice what happened to each type of term:

- $$4x^5$$ became $$20x^4$$.
- $$-3x^2$$ became $$-6x$$.
- $$7x$$ became $$7$$.
- $$-9$$ became $$0$$.

> **Exam Warning:**
> Do not subtract $$1$$ from the coefficient. In $$4x^5$$, the $$5$$ is the exponent. The derivative is $$20x^4$$, not $$3x^4$$ or $$4x^4$$.

Here are several short examples:

$$
\frac{d}{dx}(9x^7)=63x^6
\frac{d}{dx}(-2x^4)=-8x^3
\frac{d}{dx}(11x)=11
\frac{d}{dx}(-5)=0
$$

A polynomial derivative is always another polynomial. The degree usually decreases by $$1$$, unless the polynomial is constant. For example, a fifth-degree polynomial usually has a fourth-degree derivative.

> **Key Term: Degree**
> The highest power of $$x$$ with a nonzero coefficient in a polynomial.

For instance, $$p(x)=2x^6-x^3+4$$ has degree $$6$$, and

p'(x)=12x^5-3x^2
has degree $$5$$.

### Slopes and tangent lines from polynomial derivatives

The derivative of a polynomial is not just a new formula. It gives slope information about the original graph. If $$y=f(x)$$, then $$f'(a)$$ is the slope of the tangent line at $$x=a$$.

> **Key Term: Tangent line**
> A line that touches a curve at a point and has slope equal to the derivative of the function at that point, when the derivative exists.

To write a tangent line, use the point-slope form:

$$
y-y_1=m(x-x_1)
$$

where $$m=f'(a)$$ and $$(x_1,y_1)=(a,f(a))$$.

> **Question:**
> Find the equation of the tangent line to $$y=x^3-4x+1$$ at $$x=2$$.
> Solution:
> First find the point on the curve:
>
> $$
> y(2)=2^3-4(2)+1=8-8+1=1
> $$
>
> So the point is $$(2,1)$$.
> Now differentiate:
>
> $$
> y'=3x^2-4
> $$
>
> Evaluate the derivative at $$x=2$$:
>
> $$
> y'(2)=3(2)^2-4=12-4=8
> $$
>
> The tangent line has slope $$8$$ and passes through $$(2,1)$$:
>
> $$
> y-1=8(x-2)
> $$
>
> Equivalently,
>
> $$
> y=8x-15
> $$

On AP free-response questions, the unsimplified point-slope form is usually acceptable if it clearly gives the correct point and slope.

> **Test Tip:**
> For tangent-line questions, always find two things: the function value $$f(a)$$ and the derivative value $$f'(a)$$. Students often find the slope correctly but forget to compute the point on the original curve.

Polynomial derivatives also help identify where tangent lines are horizontal. A horizontal tangent has slope $$0$$, so solve $$f'(x)=0$$.

> **Key Term: Critical number**
> A number $$c$$ in the domain of $$f$$ where $$f'(c)=0$$ or where $$f'(c)$$ does not exist. For polynomial functions, derivatives exist everywhere, so critical numbers occur where $$f'(c)=0$$.
>
> **Question:**
> Find the values of $$x$$ where $$f(x)=2x^3-9x^2+12x$$ has a horizontal tangent.
> Solution:
> Differentiate:
>
> $$
> f'(x)=6x^2-18x+12
> $$
>
> A horizontal tangent occurs when $$f'(x)=0$$:
>
> $$
> 6x^2-18x+12=0
> $$
>
> Divide by $$6$$:
>
> $$
> x^2-3x+2=0
> $$
>
> Factor:
>
> $$
> (x-1)(x-2)=0
> $$
>
> Therefore,
>
> $$
> x=1 \text{ or } x=2
> $$

If the question asks for the points where the graph has horizontal tangents, not just the $$x$$-values, substitute back into the original function:

$$
f(1)=2-9+12=5
f(2)=16-36+24=4
$$

So the points would be $$(1,5)$$ and $$(2,4)$$.

### Rewriting polynomial expressions before differentiating

Some polynomial functions are not presented as simple sums of powers. You may need to expand or simplify first, especially before learning or using product and chain rules.

For example,

$$
g(x)=3(x-1)(x+2)
$$

is a polynomial, but it is not yet written term by term. Expand first:

$$
(x-1)(x+2)=x^2+x-2
$$

So

$$
g(x)=3x^2+3x-6
$$

Now differentiate:

g'(x)=6x+3
Another example:

$$
h(x)=(x-2)^3
$$

If you are applying only polynomial power rule methods, expand:

$$
(x-2)^3=x^3-6x^2+12x-8
$$

Then

h'(x)=3x^2-12x+12
Later in the course, the chain rule gives a quicker method:

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

These answers are equivalent because

$$
3(x-2)^2=3(x^2-4x+4)=3x^2-12x+12
$$

> **Exam Warning:**
> The basic power rule $$\frac{d}{dx}(x^n)=nx^{n-1}$$ applies directly to powers of $$x$$. Do not treat $$(x-2)^3$$ as if its derivative were simply $$3x^2$$. Either expand first or, when allowed, use the chain rule.

You may also need to rewrite expressions with missing powers. For example,

$$
q(x)=5x^4+2x^2-13
$$

has no $$x^3$$ or $$x$$ term, but that does not matter. Differentiate only the terms present:

q'(x)=20x^3+4x
A frequent AP-style variation asks for a derivative value rather than the whole derivative formula.

> **Question:**
> If $$p(x)=-x^4+6x^3-5x+10$$, find $$p'(1)$$.
> Solution:
> Differentiate first:
>
> $$
> p'(x)=-4x^3+18x^2-5
> $$
>
> Then substitute $$x=1$$:
>
> $$
> p'(1)=-4+18-5=9
> $$

Do not substitute into the original function when the question asks for $$p'(1)$$. The value $$p(1)$$ gives a $$y$$-coordinate, while $$p'(1)$$ gives a slope.

Polynomial derivatives also appear in motion questions. If $$s(t)$$ is position, then velocity is $$v(t)=s'(t)$$, and acceleration is $$a(t)=v'(t)=s''(t)$$.

For example, if

$$
s(t)=t^3-6t^2+9t
$$

then

$$
v(t)=3t^2-12t+9
$$

and

$$
a(t)=6t-12
$$

Even if the context uses time instead of $$x$$, the power rule works the same way. Differentiate with respect to the variable given.

## Key Point Checklist

This article has covered the following key knowledge points:

- The derivative gives instantaneous rate of change and tangent-line slope.
- The power rule states that $$\frac{d}{dx}(x^n)=nx^{n-1}$$.
- For polynomial differentiation, exponents are nonnegative integers.
- A constant term has derivative $$0$$.
- A linear term $$ax$$ has derivative $$a$$.
- The constant multiple rule lets you keep coefficients in front while differentiating.
- The sum and difference rules allow polynomial derivatives to be found term by term.
- A polynomial derivative is also a polynomial.
- To find a tangent line, compute both $$f(a)$$ and $$f'(a)$$.
- Horizontal tangents occur where $$f'(x)=0$$.
- Some polynomial expressions should be expanded before applying the power rule term by term.
- In motion problems, the derivative of position is velocity, and the derivative of velocity is acceleration.

## Key Terms and Concepts

- Derivative
- Power rule
- Polynomial
- Constant multiple rule
- Sum and difference rules
- Degree
- Tangent line
- Critical number


Applying the power rule - Application to polynomials