Derivatives — AP Calculus AB
1. Derivative as Instantaneous Rate of Change ★★☆☆☆ ⏱ 4 min
The derivative of a function $f(x)$ at $x=a$ measures the instantaneous rate of change of the function's output at that point, which equals the slope of the tangent line to the graph of $f$ at $x=a$. Common notations are $f'(a)$, $\frac{dy}{dx}\bigg|_{x=a}$, and $y'$.
Starting from the average rate of change (slope of the secant line):
\text{Average rate of change} = \frac{f(a+h) - f(a)}{h}
The limit definition of the derivative at $x=a$ is:
f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}
An equivalent alternate form is:
f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}
Exam tip: Examiners regularly test recognition of the limit definition of a derivative. If you see a limit matching the forms above, it is always the derivative of $f$ at the given point $a$.
2. Core Differentiation Rules ★★★☆☆ ⏱ 5 min
The limit definition is accurate but tedious for complex functions. These standard rules let you compute derivatives directly without evaluating limits, and are tested on every AP Calculus AB exam:
- **Power Rule**: For $f(x) = x^n$ (any real constant $n$): $\frac{d}{dx}[x^n] = nx^{n-1}$
- **Product Rule**: For $f(x) = u(x) \cdot v(x)$: $f'(x) = u'(x)v(x) + u(x)v'(x)$
- **Quotient Rule**: For $f(x) = \frac{u(x)}{v(x)}, v(x) \neq 0$: $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$
- **Chain Rule**: For composite functions $f(x) = g(h(x))$: $f'(x) = g'(h(x)) \cdot h'(x)$
Exam tip: The chain rule is the most frequently tested differentiation rule on AP Calc AB, often combined with other rules. Always check for composite functions before finalizing your derivative.
3. Implicit Differentiation ★★★☆☆ ⏱ 3 min
Explicit functions have $y$ isolated as $y = f(x)$. Implicit functions have $x$ and $y$ mixed together (e.g., a circle $x^2 + y^2 = 25$, which cannot be written as a single explicit function of $x$). Implicit differentiation lets you find $\frac{dy}{dx}$ without solving for $y$ first, following three steps:
- Differentiate both sides of the equation with respect to $x$
- Multiply by $\frac{dy}{dx}$ every time you differentiate a term containing $y$ (by the chain rule, since $y$ is a function of $x$)
- Rearrange to isolate $\frac{dy}{dx}$
Exam tip: Examiners frequently ask for the slope of a tangent line to an implicit curve at a given point. Plug in the $(x,y)$ coordinates immediately after solving for $\frac{dy}{dx}$ to avoid unnecessary algebraic simplification.
4. Derivatives of Transcendental Functions ★★★☆☆ ⏱ 3 min
These derivatives of exponential, logarithmic, and trigonometric functions are guaranteed to appear on the AP Calculus AB exam, and all rules assume angles are measured in radians:
- **Exponential**: $\frac{d}{dx}[e^x] = e^x$, $\frac{d}{dx}[e^{u(x)}] = e^{u(x)}u'(x)$
- **Logarithmic**: $\frac{d}{dx}[\ln x] = \frac{1}{x}$, $\frac{d}{dx}[\ln(u(x))] = \frac{u'(x)}{u(x)}$ for $u(x) > 0$
- **Trigonometric**: Co-functions (cos, csc, cot) have negative derivatives: $\frac{d}{dx}[\sin x] = \cos x$, $\frac{d}{dx}[\cos x] = -\sin x$, $\frac{d}{dx}[\tan x] = \sec^2 x$, $\frac{d}{dx}[\csc x] = -\csc x \cot x$, $\frac{d}{dx}[\sec x] = \sec x \tan x$, $\frac{d}{dx}[\cot x] = -\csc^2 x$
Exam tip: A common trap is forgetting the chain rule for trigonometric functions with a linear argument (e.g., $\sin(2x)$). Always multiply by the derivative of the angle term.
5. Higher-Order Derivatives ★★☆☆☆ ⏱ 2 min
You can differentiate derivatives themselves to get higher-order derivatives, which measure the rate of change of the first derivative. Common notation: 1st derivative $f'(x)$, 2nd derivative $f''(x)$, 3rd derivative $f'''(x)$, and $f^{(n)}(x)$ for $n \geq 4$. For AP Calculus AB, 2nd derivatives are most commonly used for concavity and acceleration.
6. AP-Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Students rush and only differentiate the outer function, ignoring the inner term.
Why: Students misremember the mnemonic or swap numerator terms.
Why: Students forget all calculus trig derivative rules are derived using radian measure.
Why: Students treat $y$ as a constant instead of a function of $x$.
Why: Students confuse power functions (variable base, constant exponent) with exponential functions (constant base, variable exponent).