AP Calculus AB Unit 2 Notes: Learning the Core Differentiation Toolkit

Power Rule

A huge reason derivatives become practical (instead of being a limit you re-derive every time) is that many real-world and test problems use power functions—expressions built from terms like x^n. The Power Rule gives you a fast, reliable way to differentiate these without going back to the limit definition.

What the Power Rule says (and what it means)

If n is any real number and the function is
f(x) = x^n
then its derivative is
f'(x) = n x^{n-1}
Conceptually, this rule tells you how the slope (instantaneous rate of change) of a power function depends on both:

• the exponent n (which controls how steeply the function grows or decays), and
• the current input x (because the slope changes as you move along the curve).

Why it matters

1. Polynomials are everywhere. Most modeling starts with polynomial approximations, and many AP questions embed polynomials inside larger expressions.
2. It’s the foundation for more rules. When you later learn Product/Quotient/Chain Rules, you’ll still repeatedly apply the Power Rule to pieces of expressions.
3. It connects directly to motion. If s(t) is position, then s'(t) is velocity. A common position model is a polynomial, like s(t) = t^2 or s(t) = t^3 - 4t.

How it works (intuition without redoing the full proof)

From the limit definition,
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
when you plug in f(x) = x^n, you are essentially measuring how x^n changes when you nudge x by a tiny amount h. Algebra (expanding and simplifying) leads to a pattern: one factor of n comes down, and the exponent drops by 1. You don’t need to reproduce that algebra every time—but understanding that the rule comes from “zooming in” on the curve helps it feel less like magic.

Using the Power Rule correctly

To use it, you:

1. Identify the exponent on x.
2. Multiply by that exponent.
3. Subtract 1 from the exponent.

This works for many kinds of exponents, not just positive integers.

Example 1: Basic polynomial term

Differentiate
f(x) = x^7
Apply the rule with n = 7:
f'(x) = 7x^6

Example 2: Negative exponent (reciprocal-style)

Differentiate
g(x) = x^{-3}
Apply the rule with n = -3:
g'(x) = -3x^{-4}
You can rewrite to make the meaning clearer:
g'(x) = -\frac{3}{x^4}
A common conceptual check: x^{-3} = \frac{1}{x^3} decreases for positive x, so a negative derivative makes sense.

Example 3: Fractional exponent (root-style)

Differentiate
h(x) = x^{1/2}
Apply the rule with n = \frac{1}{2}:
h'(x) = \frac{1}{2}x^{-1/2}
Rewrite if desired:
h'(x) = \frac{1}{2\sqrt{x}}
This also highlights a domain issue: the derivative requires x > 0 if you want real outputs.

What goes wrong most often

A frequent error is to change the exponent incorrectly (students sometimes subtract 1 from the coefficient instead of the exponent). Another is mishandling negative exponents, especially when rewriting as fractions.

Exam Focus

• Typical question patterns
  • Differentiate a polynomial like 3x^5 - 2x^2 + 9.
  • Differentiate expressions with negative or fractional exponents (often written using radicals or reciprocals).
  • Use the derivative to evaluate a slope at a point, such as finding f'(2).
• Common mistakes
  • Dropping the exponent to n-1 but forgetting to multiply by n.
  • Turning x^{-4} into -\frac{1}{x^4} (the negative sign is not automatic—it comes only from the coefficient).
  • Treating \sqrt{x} as if it differentiates to \sqrt{1}; you must rewrite \sqrt{x} = x^{1/2} (or know the derivative form).

Derivative Rules: Constant, Sum, Difference, Constant Multiple

Once you can differentiate single power terms, the next step is learning how derivatives behave with combinations of functions. These rules are sometimes grouped as linearity rules because the derivative operator distributes over addition/subtraction and pulls out constant multiples.

What these rules say

Suppose f(x) and g(x) are differentiable and c is a constant.

Constant Rule: The derivative of a constant is zero.
\frac{d}{dx}(c) = 0
This matches the idea of slope: a horizontal line has slope 0 everywhere.

Constant Multiple Rule: You can pull constant factors out of a derivative.
\frac{d}{dx}(c f(x)) = c f'(x)

Sum Rule: The derivative of a sum is the sum of the derivatives.
\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)

Difference Rule: The derivative of a difference is the difference of the derivatives.
\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)

Why they matter

These rules are what make derivatives scalable. Real expressions are almost never single-term; they’re built from multiple pieces. Linearity rules let you differentiate term-by-term instead of treating the whole expression as a complicated unit.

A useful analogy: think of the derivative as a “rate-of-change machine.” If you scale a quantity by 5, its rate of change scales by 5 (constant multiple). If you add two quantities, their total rate of change is the sum of their rates (sum rule).

How to apply them (process you can repeat)

When you see an expression like a polynomial, you can:

1. Break it into separate terms using sum/difference.
2. Pull out any constant coefficients.
3. Apply the Power Rule to each variable power term.
4. Simplify.

Notation reference (same idea, different symbols)

AP problems use multiple derivative notations. They mean the same derivative but appear in different contexts.

Worked examples

Example 1: Differentiate a polynomial

Find the derivative of
f(x) = 3x^5 - 2x^2 + 9
Use constant multiple and sum/difference rules to differentiate term-by-term:

• \frac{d}{dx}(3x^5) = 3\cdot 5x^4 = 15x^4
• \frac{d}{dx}(-2x^2) = -2\cdot 2x = -4x
• \frac{d}{dx}(9) = 0
  So,
  f'(x) = 15x^4 - 4x

Example 2: Evaluate the derivative at a point

Let
g(x) = -\frac{4}{x^3} + 2x
Rewrite using exponents:
g(x) = -4x^{-3} + 2x
Differentiate:
g'(x) = -4(-3)x^{-4} + 2 = 12x^{-4} + 2
Evaluate at x = 2:
g'(2) = 12\cdot 2^{-4} + 2 = 12\cdot \frac{1}{16} + 2 = \frac{3}{4} + 2 = \frac{11}{4}
Interpreting this: the slope of the tangent line at x = 2 is \frac{11}{4}.

What goes wrong most often

Students often try to apply the Power Rule to a sum “all at once,” like treating x^2 + x as if it were x^3. The sum rule is what prevents that—you must differentiate each term separately.

Another common slip is losing negative signs during the difference rule. A good habit is to keep parentheses around terms with negatives until the end.

Exam Focus

• Typical question patterns
  • Differentiate polynomials and simplify.
  • Find the slope of a tangent line by computing f'(a).
  • Use derivative notation flexibly: compute \frac{dy}{dx} given y = \text{(expression)}.
• Common mistakes
  • Differentiating a constant incorrectly (e.g., thinking \frac{d}{dx}(9) = 9).
  • Dropping a negative sign when differentiating something like -2x^2.
  • Trying to combine unlike terms incorrectly after differentiating (for example, combining 15x^4 and -4x).

Derivatives of Trigonometric Functions

Trigonometric functions are essential in calculus because they model periodic behavior: sound waves, alternating current, seasonal temperature changes, circular motion, and more. In AP Calculus AB, you’re expected to know the basic trig derivatives and use them fluently.

Angle measure matters (radians vs degrees)

A crucial fact: the standard derivative formulas for trig functions assume the input is in radians, not degrees. In calculus, radians are the natural unit because they connect angle measure directly to arc length on a unit circle, which is why the derivative relationships come out clean.

If you used degrees inside trig functions, the derivatives would include extra conversion factors. On the AP exam, you should assume trig inputs are in radians unless explicitly stated otherwise.

The core trig derivative facts

The two most fundamental trig derivatives are:
\frac{d}{dx}(\sin x) = \cos x
\frac{d}{dx}(\cos x) = -\sin x
These capture a deep geometric relationship: as you move around the unit circle, the rate of change of the y-coordinate (sine) is the x-coordinate (cosine), and the rate of change of cosine is negative sine.

From these, you also commonly use:
\frac{d}{dx}(\tan x) = \sec^2 x
And for completeness (also commonly used in AP Calculus):
\frac{d}{dx}(\sec x) = \sec x \tan x
\frac{d}{dx}(\csc x) = -\csc x \cot x
\frac{d}{dx}(\cot x) = -\csc^2 x

Why these derivatives matter

1. Modeling oscillations: If a position function involves \sin t or \cos t, the velocity and acceleration naturally involve trig derivatives.
2. Tangent line and slope problems: Trig graphs change direction frequently, so derivative sign and value questions are very common.
3. Building blocks: Later, when you differentiate more complex functions (products, compositions), these basic trig derivatives are the pieces you’ll need.

How to use trig derivatives correctly

The main skill is pattern recognition: identify the trig function and apply its derivative formula, then use linearity rules to handle sums and constants.

Be careful about two things:

• The negative sign in \frac{d}{dx}(\cos x) = -\sin x.
• Domain restrictions: \tan x and \sec x are undefined where \cos x = 0; \csc x and \cot x are undefined where \sin x = 0. The derivative will share these restrictions.

Worked examples

Example 1: Differentiate a trig combination

Differentiate
f(x) = 4\sin x - 3\cos x
Apply constant multiple and sum/difference rules:
f'(x) = 4\cos x - 3(-\sin x)
So,
f'(x) = 4\cos x + 3\sin x

Example 2: Tangent slope at a point

Let
g(x) = \cos x + \tan x
Then
g'(x) = -\sin x + \sec^2 x
If you want the slope at x = 0:

• \sin 0 = 0
• \sec 0 = 1 so \sec^2 0 = 1
  Thus,
  g'(0) = -0 + 1 = 1
  So the tangent line slope at x = 0 is 1.

What goes wrong most often

• Forgetting radians: Students sometimes think this is a minor technicality, but the derivative formulas are built around radian measure.
• Sign errors: The negative in the cosine derivative is one of the most tested “tiny details.”
• Confusing related functions: For example, mixing up \frac{d}{dx}(\tan x) with \sec x \tan x (that’s for \sec x, not \tan x).

Exam Focus

• Typical question patterns
  • Differentiate sums of trig functions and evaluate at a point.
  • Find where a trig function is increasing/decreasing using the sign of the derivative.
  • Compute the slope of a tangent line to a trig graph at a specified x-value.
• Common mistakes
  • Writing \frac{d}{dx}(\cos x) = \sin x (missing the negative sign).
  • Using \frac{d}{dx}(\tan x) = \sec x \tan x (confusing tan with sec).
  • Ignoring where the original trig function is undefined, then reporting derivative values at those points.

Derivatives of Exponential and Logarithmic Functions

Exponential and logarithmic functions are the calculus language of growth, decay, and “per-unit” change. They show up in population models, radioactive decay, continuously compounded interest, and any situation where the rate of change depends on the current amount.

Exponential functions and why e is special

An exponential function has the variable in the exponent, like a^x. Among all bases, the number e (Euler’s number) is special because it makes the derivative of e^x exactly itself:
\frac{d}{dx}(e^x) = e^x
This is not just a neat fact—it’s why e^x is used for continuous growth/decay. If a quantity’s rate of change is proportional to its current size, exponential functions appear naturally.

For a general base a > 0 with a \ne 1:
\frac{d}{dx}(a^x) = a^x \ln(a)
So the derivative is still proportional to a^x, but scaled by \ln(a).

Logarithmic functions as “undoing” exponentials

A logarithm answers “what exponent gives this value?” For base e, the natural logarithm is \ln x, which is the inverse of e^x.

The core derivative you use in AP Calculus AB is:
\frac{d}{dx}(\ln x) = \frac{1}{x}
This derivative is one reason logarithms are so useful: they turn multiplicative growth into additive structure (a big idea later in calculus, especially with logarithmic differentiation).

For other bases:
\frac{d}{dx}(\log_a x) = \frac{1}{x \ln(a)}
where a > 0 and a \ne 1.

Domains you must respect

• \ln x and \log_a x are defined (in real numbers) only for x > 0.
• \frac{1}{x} is undefined at x = 0.
  So any derivative involving logs inherits these restrictions. On exam problems, it’s common to be asked about behavior near x = 0 or to evaluate derivatives only at valid points.

How to apply these derivatives

At the “basic rules” stage, most exponential/log derivative problems rely on:

• recognizing whether you have e^x, a^x, \ln x, or \log_a x,
• applying the correct formula, and
• using constant multiple and sum/difference rules.

(If the exponent or the log input is more complicated than just x, that typically calls for the Chain Rule, which is a later rule—even though it often gets paired with these derivatives in practice.)

Worked examples

Example 1: Differentiate an expression with e^x

Differentiate
f(x) = 7e^x - 5
Use constant multiple and constant rule:
f'(x) = 7e^x - 0
So,
f'(x) = 7e^x

Example 2: Differentiate a general exponential

Differentiate
g(x) = 2^x
Use the base-a exponential derivative:
g'(x) = 2^x \ln(2)
A common interpretation: the slope of 2^x at any point is proportional to its current value, scaled by \ln(2).

Example 3: Differentiate a logarithm

Differentiate
h(x) = \ln x + 3x^2
Differentiate term-by-term:
h'(x) = \frac{1}{x} + 6x
This derivative makes sense dimensionally: \ln x changes slowly for large x, and \frac{1}{x} reflects that slowing rate.

Example 4: Base-10 style log derivative

Differentiate
p(x) = \log_a x
The derivative is
p'(x) = \frac{1}{x \ln(a)}
If you see \log x without a base, the base depends on context (often base 10 in some settings). On the AP exam, if a base matters, it’s typically specified; otherwise, many calculus expressions use \ln explicitly.

What goes wrong most often

• Forgetting the \ln(a) factor when differentiating a^x.
• Treating \ln x like a power (it is not x^{-1}; rather, its derivative is \frac{1}{x}).
• Ignoring domain and evaluating a log derivative at non-positive x.

Exam Focus

• Typical question patterns
  • Differentiate expressions involving e^x, a^x, and constants.
  • Differentiate sums like e^x + \ln x and evaluate at a specific x value.
  • Interpret derivatives of exponential models as growth/decay rates (often in applied word problems).
• Common mistakes
  • Writing \frac{d}{dx}(2^x) = 2^x (true only for base e, not base 2).
  • Writing \frac{d}{dx}(\ln x) = \ln(x-1) or other “inside changes” (that confusion usually comes from mixing up derivative rules with function transformations).
  • Plugging in x = 0 or negative values when evaluating derivatives involving \ln x.