AP Calculus AB Unit 2: Basic Differentiation Rules (Power, Linearity, Trig, Exponential, Logarithmic)

Derivative

Measures how fast a function’s output changes when the input changes; geometrically, it is the slope of the tangent line at a point.

Basic Differentiation Rules

Efficient algebraic shortcuts for finding derivatives that match the limit definition.

Workflow for differentiation

Rewrite → Differentiate → Simplify; a process that can prevent errors.

Limit definition of derivative

f'(x)=lim(h→0) [f(x+h) - f(x)]/h.

Power Rule

If f(x) = x^n, then f'(x) = nx^(n-1).

Constant Rule

The derivative of a constant function f(x) = c is f'(x) = 0.

Constant Multiple Rule

If a function is multiplied by a constant c, then the derivative is c * f'(x).

Sum Rule

If F(x) = f(x) + g(x), then F'(x) = f'(x) + g'(x).

Difference Rule

If F(x) = f(x) - g(x), then F'(x) = f'(x) - g'(x).

Negative Exponent

A negative exponent indicates a reciprocal (e.g., x^-n = 1/x^n).

Radical expression as exponent

A square root can be rewritten as x^(1/2) for easier differentiation.

Derivative of sin(x)

The derivative of sin x is cos x.

Derivative of cos(x)

The derivative of cos x is -sin x.

Derivative of tan(x)

The derivative of tan x is sec^2 x.

Exponential function derivative

If f(x) = e^x, then f'(x) = e^x.

Logarithmic derivative

If f(x) = ln(x), then f'(x) = 1/x for x > 0.

General logarithmic rule

If f(x) = log_a(x), then f'(x) = 1/(x ln a) for x > 0.

Linearity in differentiation

Differentiation distributes over addition and subtraction.

Chain Rule

A method for differentiating composite functions; applies when one function is nested inside another.

Mistake: Loss of negative sign

Forgetting that the derivative of cos(x) is -sin(x).

Derivative of a constant term

The derivative of any constant is zero.

Writing incorrect implications

Improper notation such as claiming y=x^2 = 2x rather than y=x^2 ⇒ y' = 2x.

Common error with radicals

Failing to rewrite a radical before differentiating.

Exponential growth rate

The property that the derivative of e^x is proportional to itself (e^x).

Memory aid for trigonometric derivatives

The 'PSST' mnemonic: Tangent goes with Secant, Cotangent goes with Cosecant.

F'(x) evaluated at a point

To evaluate, first find the general derivative F'(x), then substitute the point.

Improper logarithm use

Assuming log properties allow splits of logs incorrectly.

Typical exam focus

Differentiating polynomials, exponential, trigonometric, and logarithmic functions.