The derivative of a function is defined as a limit of a difference quotient. The two common forms are: - At a general x: f′(x) = lim_{h→0} [f(x+h) − f(x)] / h (if this limit exists)—this gives the derivative function. - At a specific point a: f′(a) = lim_{x→a} [f(x) − f(a)] / (x − a) (equivalent form). Notation you’ll see on the AP: f′(x), dy/dx, or y′. Conceptually, this limit gives the instantaneous rate of change of f at x and is the slope of the tangent line there (useful for CHA-2.B and CHA-2.C tasks on the AP). For tangent-line equation at x = a: y = f(a) + f′(a)(x − a). For more AP-aligned review, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the limit definition: f'(x) = lim_{h→0} [f(x+h) − f(x)]/h. Step-by-step for a concrete f (say f(x)=x^2): 1. Plug into the difference quotient: [ (x+h)^2 − x^2 ] / h. 2. Expand and simplify numerator: (x^2 + 2xh + h^2 − x^2)/h = (2xh + h^2)/h. 3. Factor h: h(2x + h)/h. 4. Cancel h (for h ≠ 0): 2x + h. 5. Take the limit as h → 0: lim_{h→0} (2x + h) = 2x. So f'(x)=2x. Key tips tied to the CED: always simplify algebraically before taking the limit (CHA-2.B.2). If the left and right limits as h→0 disagree, the derivative doesn't exist (differentiability at a point, CHA-2.C.1). The derivative at x gives the tangent slope—use it to write the tangent line: y = f(a) + f'(a)(x−a). For more worked examples and AP-style practice, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and the Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

Use whichever notation fits the context—they all mean “the derivative,” but each is more convenient in different situations (CED CHA-2.B.3). - f′(x): best when you think of the derivative as a function of x (analytic work, applying derivative rules, writing f′(x)=limit of the difference quotient). Clean for formulas and graph/analysis tasks (CHA-2.B.2, CHA-2.B.4). - dy/dx: Leibniz form is great when variables and rates matter (related rates, differentials, separation of variables, or emphasizing “rate of change of y with respect to x”). It makes chain rule and implicit differentiation look natural. - y′ (or f′): short and handy when y = f(x) and you’re working pointwise (tangent-line slope at a point) or doing quick algebraic steps. Examples: slope of tangent at x = a → use f′(a) or (dy/dx)|_{x=a}; implicit differentiation → use dy/dx; concise manipulation in proofs → y′ or f′(x). For higher derivatives use f′′(x) or d²y/dx² as appropriate. For review and practice, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2). For extra practice problems, go to (https://library.fiveable.me/practice/ap-calculus).

The derivative at a point is a single number: the limit of the difference quotient at x = a, f′(a) = lim_{h→0} [f(a+h)-f(a)]/h (when it exists). It’s the instantaneous rate of change there and the slope of the tangent line at that one x-value. The derivative function, written f′(x) or dy/dx, is a new function that gives those pointwise slopes for every x where the limit exists (CHA-2.B.2, CHA-2.C.1). So: f′(2) is one slope; f′(x) is a rule you can evaluate to get slopes at any x. If f′(a) doesn’t exist, the derivative function is undefined at a (non-differentiability). This distinction appears often on the AP exam—be ready to use the limit definition, graph/table/analytic representations, and to write tangent-line equations (see Topic 2.2 study guide: https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD; unit overview: https://library.fiveable.me/ap-calculus/unit-2). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).

The difference quotient is just the slope of a secant line through (x, f(x)) and (x+h, f(x+h)). Set it up as (f(x+h) − f(x)) / h, then take the limit as h → 0 to get the derivative f′(x) (CHA-2.B.2). Steps to use it: 1. Write f(x+h). 2. Form f(x+h) − f(x). 3. Divide that difference by h. 4. Algebraically simplify (cancel h factors). 5. Take the limit as h → 0. Quick example: f(x)=x^2. - f(x+h)=(x+h)^2 = x^2 + 2xh + h^2 - Difference: (x^2+2xh+h^2) − x^2 = 2xh + h^2 - Divide by h: (2xh + h^2)/h = 2x + h - Limit as h→0: f′(x)=2x Remember: if you can’t cancel h, the limit may not exist (not differentiable). For AP review and more examples see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

1) Get the point: you need (a, f(a)). 2) Find the slope of the tangent = f′(a). Either - Use the limit (difference quotient): f′(a) = lim_{h→0} [f(a+h)−f(a)]/h (CED CHA-2.B), or - Use derivative rules (power, product, chain, etc.) and then evaluate at x = a (CED CHA-2.B.3). 3) Write the line in point-slope form: y − f(a) = f′(a)(x − a). That’s the tangent line. Quick notes: if you have parametric equations, slope = (dy/dt)/(dx/dt) at the given t. If f′(a) doesn’t exist, there’s no tangent line there (not differentiable). On the AP exam show work: either the limit computation or the derivative rule you used and the evaluation, then the point-slope step. For more examples and step-by-step practice, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and extra problems (https://library.fiveable.me/practice/ap-calculus).

Think of the difference quotient, (f(x+h) − f(x))/h, as the slope of a secant line through the points (x, f(x)) and (x+h, f(x+h)). As h gets smaller those two points get closer, so the secant line “rotates” and approaches one limiting line—the tangent at x. The limit as h → 0 picks out that limiting slope (if it exists), which is the instantaneous rate of change of f at x. That limiting value is the derivative f′(x) (CHA-2.B.2) and, by definition, is the slope of the tangent line at that point (CHA-2.C.1). Geometrically: secants → tangent; analytically: difference quotients → derivative. For more practice and the AP framing, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) or Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2).

Saying “the derivative doesn’t exist at a point” means the limit that defines the derivative, lim_{h→0} [f(x+h)-f(x)]/h, does not exist (so f′(x) is not a real number). That can happen for a few AP-tested reasons: the two one-sided difference-quotient limits aren’t equal (a corner), one or both blow up to ±∞ (a vertical tangent), or f isn’t even continuous there (so the derivative can’t exist). Graphically, there’s no single tangent line with a well-defined slope at that x. On the exam you’ll often justify nondifferentiability by showing the limit fails or by pointing to discontinuity, corner, or vertical tangent (CED CHA-2.B and CHA-2.C). For a quick review, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try practice problems at (https://library.fiveable.me/practice/ap-calculus).

Start with the limit definition (CHA-2.B.2): f′(x) = lim_{h→0} [f(x+h) − f(x)]/h. For f(x)=x^2: 1. f(x+h) = (x+h)^2 = x^2 + 2xh + h^2. 2. Difference quotient: [(x^2 + 2xh + h^2) − x^2]/h = (2xh + h^2)/h = 2x + h. 3. Take the limit as h → 0: f′(x) = lim_{h→0} (2x + h) = 2x. So the derivative function is f′(x)=2x (also written dy/dx or y′). That value at a specific x gives the slope of the tangent line there (CHA-2.C.1). Example: at x=3, slope = 2(3)=6, so tangent line is y − 9 = 6(x − 3). If you want a quick review of this topic, check the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Short answer: use the limit definition when the problem specifically asks for the derivative as a limit or you need to show differentiability from first principles (CHA-2.B). Use derivative rules (power, constant multiple, sum/difference, product, quotient, chain) for speed and efficiency on routine analytic problems and most AP exam items. When to pick which: - Limit definition (f(x+h)-f(x))/h as h→0—use this to - represent f′(x) as a limit (CED CHA-2.B.2), - prove a derivative at a tricky point or show non-differentiability, - find the slope of a tangent when no rule applies. - Derivative rules—use these whenever f is built from standard pieces (polynomials, trig, exp, logs, compositions). The AP exam expects you to apply these rules quickly in multiple-choice and free-response. Tip: you should be able to do a short limit proof for at least one function (CED LO CHA-2.B) but rely on rules for most work. Review Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and drill problems at https://library.fiveable.me/practice/ap-calculus to build both skills.

Step-by-step you can always follow this 3-part recipe. 1) Find the slope (the derivative at the point). If the point is (a, f(a)), compute m = f′(a). Use the limit definition m = lim_{h→0} [f(a+h)-f(a)]/h if you must (CHA-2.B), or use derivative rules if you can. 2) Use point-slope form of the line: y − f(a) = m(x − a). (Remember f′(a) is the slope of the tangent—CHA-2.C.1.) 3) Simplify to your preferred form. Quick example: f(x)=x^2 at a=3. f′(x)=2x so m=f′(3)=6. Point (3,9). Tangent: y−9=6(x−3) → y=6x−9. For AP: show the derivative step (limit or rule) and units/interpretation if context appears (they ask for interpretation on the exam). For extra practice and examples see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

A secant line through (x, f(x)) and (x+h, f(x+h)) has slope [f(x+h) − f(x)]/h—that’s the difference quotient (average rate of change). The derivative at x, f′(x), is the limit of those secant slopes as h → 0: f′(x) = lim_{h→0} [f(x+h) − f(x)]/h (CHA-2.B.2). Geometrically, as the second point moves closer (h gets smaller), the secant line “rotates” and approaches the tangent line; the tangent’s slope is the instantaneous rate of change (CHA-2.C.1). If that limit exists, the function is differentiable at x; if secant slopes don’t settle to one value, no derivative there. Use f′(x) as the slope in the tangent-line equation y − f(x) = f′(x)(x − x0). For more AP-aligned practice and examples on limits of difference quotients, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Short answer: both are correct—use whichever fits the context and makes your reasoning clearest. How to choose: - If the problem gives a named function f(x) and asks for “the derivative as a function,” write f′(x). (CED CHA-2.B.3) - If you need the derivative at a specific x, either f′(3) or (dy/dx)|_{x=3} is fine; be consistent. - Use dy/dx (Leibniz) when doing implicit differentiation, related-rates, or when you want to emphasize the variable with respect to which you’re differentiating (e.g., dy/dt, dx/dt). Leibniz helps you track dependent/independent variables in rates problems. - Use f′(x) or y′ for concise algebraic work or when the function is named y = f(x). On the AP exam, clear notation counts (Practice 4: Communication and Notation), so pick the form that makes your steps easiest to follow and switch only when it helps (and indicate what you mean). For a quick refresher, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

If you’re getting different answers, one of these is almost always the cause: - Algebra mistakes when you simplify (expand, factor, cancel). The difference quotient needs correct algebra before you take the limit—don’t plug h = 0 too early. - Cancelling h incorrectly. For rational or root expressions you usually factor or multiply by a conjugate so an h cancels; if it doesn’t cancel the limit might not exist. - Using the wrong limit form. f'(x) = lim(h→0)[f(x+h)-f(x)]/h versus the point form lim(x→a)[f(x)-f(a)]/(x-a). They’re equivalent, but mixing steps can cause errors. - One-sided limits differ (corner, cusp, vertical tangent) → derivative doesn’t exist, so different one-sided approaches give different values. - Trig or exponential functions need identities (e.g., sin addition formula)—missing that gives wrong limits. Checklist: do algebra symbolically, cancel h, then take h→0; verify both one-sided limits at a point; if you still disagree, plug into a graphing calculator or compute the derivative with rules to compare. For worked examples and practice, see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2).

Graphically: f′(x) is the slope of the tangent line at each x. On a graph of f, you estimate slope by drawing tangents or by seeing where secant slopes approach a limit; on a graph of f′ you read sign (increasing/decreasing), zeros (critical points), and concavity (f′ increasing/decreasing). This matches CHA-2.C.1 and CHA-2.B.4. Numerically: you use tables or difference quotients. Approximate f′(a) with (f(a+h)−f(a))/h for small h or use symmetric/forward differences from data. That’s how the exam asks you to “estimate” derivatives from tables (Topic 2.3 connection). Analytically: you use the limit definition f′(x)=lim_{h→0}(f(x+h)−f(x))/h or derivative rules to find an exact formula f′(x) (CHA-2.B.2, CHA-2.B.3). From that you get the tangent-line equation and exact rates. On the AP exam you’ll be asked to connect these (interpret graphs/tables, compute limits, write tangent lines). For a focused review see the Topic 2.2 study guide (https://library.fiveable.me/ap-calculus/unit-2/defining-derivative-function-using-derivative-notation/study-guide/j9KaEWbB5OECijykhxCD) and try practice problems (https://library.fiveable.me/practice/ap-calculus).