d/dx[sin x] = cos x, and d/dx[cos x] = −sin x. You can get these from the limit definition of the derivative using the standard limits lim_{h→0} (sin h)/h = 1 and lim_{h→0} (1−cos h)/h = 0 (this ties to the CED’s LIM-3.A learning objective). Remember the chain rule: if u = u(x), then d/dx[sin(u)] = cos(u)·u′ and d/dx[cos(u)] = −sin(u)·u′. Higher-order derivatives of sin and cos cycle every 4 derivatives (sin → cos → −sin → −cos → sin). These rules are essential for Topic 2.7 (FUN-3.A.4) on the AP Calc CED and show up often on free-response and multiple-choice items. For a focused review, check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and more unit practice at (https://library.fiveable.me/ap-calculus/unit-2). For lots of practice problems, see (https://library.fiveable.me/practice/ap-calculus).

Use the limit definition of the derivative. For f(x) = e^x, f'(x) = lim_{h→0} (e^{x+h} − e^x)/h = lim_{h→0} e^x (e^h − 1)/h = e^x · lim_{h→0} (e^h − 1)/h. So you just need lim_{h→0} (e^h − 1)/h = 1. One easy justification: use the Maclaurin series for e^h: e^h = 1 + h + h^2/2! + … . Then (e^h − 1)/h = 1 + h/2! + … → 1 as h→0. Therefore f'(x) = e^x. Also remember the chain-rule version: d/dx[e^{u(x)}] = e^{u(x)}·u'(x). This matches the AP CED keywords (natural exponential, derivative as limit, chain rule). For extra review see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv), the Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2), and practice problems (https://library.fiveable.me/practice/ap-calculus).

The derivative of ln x is 1/x (not x). Reason: ln x is the natural logarithm defined for x>0, and by the limit/definition or by differentiating the inverse of e^x, d/dx[ln x] = 1/x for x>0. More generally, d/dx[ln|x|] = 1/x for x≠0. Quick check using the chain rule: if u(x) is a differentiable function, d/dx[ln(u(x))] = u'(x)/u(x) (so you divide the inner derivative by u). That’s an important rule on the AP CED (FUN-3.A and using inverse/chain ideas in LIM-3.A). If you want more examples and AP-style practice on this and derivatives of sin, cos, e^x, see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and thousands of practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the chain rule whenever the function you’re differentiating is a composition—that is, when “inside” the basic function isn’t just x. The basic derivatives from Topic 2.7 are for the simple forms: - d/dx[sin x] = cos x - d/dx[cos x] = −sin x - d/dx[e^x] = e^x - d/dx[ln x] = 1/x If there’s an inner function u(x), apply chain rule: d/dx[f(u(x))] = f′(u(x)) · u′(x). Examples: - d/dx[sin(3x)] = cos(3x)·3 - d/dx[cos(x^2)] = −sin(x^2)·2x - d/dx[e^{x^2}] = e^{x^2}·2x - d/dx[ln(3x+1)] = 1/(3x+1)·3 Also use it for things like (sin x)^2: treat as f(u) with f(u)=u^2, u=sin x → derivative = 2 sin x · cos x. Chain rule is part of the composite-function skills tested on the AP (see Unit 3 connections). For a quick review, check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and more practice problems (https://library.fiveable.me/practice/ap-calculus).

d/dx[sin x] = cos x. That’s the basic derivative from the CED (FUN-3.A): sine’s rate of change at x is cos x. d/dx[sin(2x)] uses the chain rule because the inside is a function (2x). Differentiate the outer sin → cos(2x), then multiply by derivative of the inside (2). So d/dx[sin(2x)] = 2·cos(2x). Key idea: sin(2x) changes twice as fast horizontally as sin x, so its slope picks up that factor 2. You can also see this from the limit definition of derivative by substituting the inner function (LIM-3.A.1). If you want practice problems and a short study guide on these rules (including limits like sin x / x), check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and more practice at (https://library.fiveable.me/practice/ap-calculus).

Whenever you see e^(something), use the chain rule: if u(x) = the “something,” then d/dx [e^{u(x)}] = e^{u(x)} · u'(x). In words: the derivative of e^u is e^u times the derivative of the inside function. Quick steps: 1. Identify u(x). 2. Differentiate u(x) to get u'(x). 3. Multiply e^{u(x)} by u'(x). Examples: - d/dx[e^{3x}] = e^{3x}·3 = 3e^{3x}. - d/dx[e^{x^2}] = e^{x^2}·2x = 2x e^{x^2}. - If f(x)=e^{x}\sin x, use product rule then chain for e^x (here u(x)=x so u'=1): f'(x)=e^x·sin x + e^x·cos x. On the AP exam this is FUN-3.A and uses the chain rule (CED keywords). For extra practice and topic review, see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and hundreds of practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of the derivative as the limit f′(x) = lim_{h→0} [cos(x+h) − cos x]/h. Use the trig identity cos(x+h) = cos x cos h − sin x sin h, so [cos(x+h) − cos x]/h = cos x · (cos h − 1)/h − sin x · (sin h)/h. As h → 0 we use the standard limits lim_{h→0} (sin h)/h = 1 and lim_{h→0} (cos h − 1)/h = 0 (these come from unit-circle geometry and the limit-definition work in the CED, LIM-3.A). That leaves f′(x) = cos x·0 − sin x·1 = −sin x. So the negative sign comes from the identity expansion: cosine’s change picks up a −sin x factor because the sin term appears with a minus in the expansion. This derivation is exactly the kind of limit-definition reasoning AP expects (Topic 2.7); for more review see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Think of a limit as “the derivative hidden in disguise.” Example: evaluate L = lim_{h→0} [sin(x+h) − sin x]/h. This is exactly the definition of f′(x) for f(x)=sin x, so L = cos x (by the known derivative). If you want to see it from raw limits, use the sine addition formula: [sin(x+h) − sin x]/h = [sin x cos h + cos x sin h − sin x]/h = sin x[(cos h − 1)/h] + cos x[sin h/h]. As h→0, sin h/h → 1 and (cos h − 1)/h → 0, so the limit = sin x·0 + cos x·1 = cos x. This uses the fundamental limit sin u/u → 1 and the derivative-as-limit idea (LIM-3.A in the CED). For more examples (like e^x and ln x) see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and extra practice problems (https://library.fiveable.me/practice/ap-calculus).

For the natural logarithm ln x the basic derivative is - d/dx [ln x] = 1/x for x > 0. For a composition (chain rule): if u(x) is a differentiable function with u(x) > 0, then - d/dx [ln(u(x))] = u′(x) / u(x). Notes that matter for AP: the 1/x rule follows from the inverse relationship with e^x and the limit definition of derivative (ties to LIM-3.A in the CED). Always watch domain—ln is only defined for positive inputs, so your derivative applies where u(x) > 0. This rule is frequently used on the exam for logarithmic differentiation and to simplify limits and derivatives (Topic 2.7, FUN-3.A and use of chain rule). For a quick refresher, check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and try extra problems at (https://library.fiveable.me/practice/ap-calculus).

Use the basic formulas whenever your function is exactly one of those basic functions (d/dx[sin x]=cos x, d/dx[cos x]=−sin x, d/dx[e^x]=e^x, d/dx[ln x]=1/x). Use the product rule only when your function is the product of two (or more) functions of x. Product rule: (f·g)' = f'·g + f·g'. Quick checks: - If you see sin(x) or e^x alone → use the basic rule. - If you see x·sin x or (2x+1)·e^{x} → product rule (differentiate each factor and add). - If you see sin(x^2) or e^{3x} → chain rule, not product rule: (sin(u))' = cos(u)·u'. Examples: - d/dx[x·sin x] = 1·sin x + x·cos x (product rule). - d/dx[sin(x^2)] = cos(x^2)·2x (chain rule). For more AP-aligned examples and practice, see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and try problems at (https://library.fiveable.me/practice/ap-calculus).

Most likely your calculator is giving a different answer because of one of two common issues: - You used a base-10 log key (log) instead of the natural log key (ln). d/dx[ln(3x)] = (3)/(3x) = 1/x. But d/dx[log10(3x)] = 1/(x·ln 10), so those answers look different—check that you pressed the ln button. - Or you asked the calculator to numerically approximate the derivative (a difference quotient) with a finite step size, which can give a slightly different decimal value. Exact symbolic differentiation uses the chain rule: if u = 3x, d/dx[ln u] = u′/u = 3/(3x) = 1/x (for x > 0). On the AP exam you’re expected to apply the chain rule and know ln means natural log (Topic 2.7, FUN-3.A.4). If you want step-by-step review, see the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv). For extra practice problems, check (https://library.fiveable.me/practice/ap-calculus).

You’ll want to memorize these basic derivatives and a couple quick consequences: - d/dx [sin x] = cos x - d/dx [cos x] = −sin x - d/dx [e^x] = e^x (for natural exponential) - d/dx [ln x] = 1/x (x > 0) Common extensions you should also know: - Chain rule: d/dx [f(g(x))] = f′(g(x))·g′(x). So d/dx [sin(3x)] = cos(3x)·3. - For a^x (a > 0): d/dx [a^x] = a^x ln a (remember e^x is the special case). - Higher-order trig pattern: derivatives of sin, cos cycle every 4 steps (sin → cos → −sin → −cos → sin...). - Key limits used in proofs: lim_{x→0} (sin x)/x = 1 and lim_{x→0} (1−cos x)/x = 0 (used to get trig derivatives from the limit definition). These are tested directly and used constantly on AP problems in Topic 2.7 (FUN-3.A, LIM-3.A). For a quick Topic 2.7 review, check the study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv). For broader Unit 2 review and lots of practice problems, see (https://library.fiveable.me/ap-calculus/unit-2) and (https://library.fiveable.me/practice/ap-calculus).

Use the product rule. If f(x) = e^x·sin x, then f′(x) = (e^x)·(sin x)′ + (e^x)′·(sin x) = e^x·cos x + e^x·sin x = e^x[sin x + cos x]. Why that works: the product rule (from Topic 2 Differentiation) says (uv)′ = u′v + uv′. You also use the basic derivatives from Topic 2.7: (e^x)′ = e^x and (sin x)′ = cos x (FUN-3.A in the CED). On the AP exam you’ll often combine these rules, so practice recognizing when to apply product, chain, and basic trig/exponential rules. Want more worked examples and AP-style practice for this topic? Check the Topic 2.7 study guide on Fiveable (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Step-by-step: identify the structure, pick the right rule, then simplify. 1) Know the basic derivatives (FUN-3.A.4): d/dx sin x = cos x; d/dx cos x = −sin x; d/dx e^x = e^x; d/dx ln x = 1/x. 2) Use linearity first: derivative of a sum/difference and constant multiples: d(ax + b f(x)) = a f′(x). 3) If you see a product, use product rule: (uv)' = u'v + uv'. If a quotient, use quotient rule or rewrite. 4) If a function is “inside” another (e.g., sin(3x), e^{2x}, ln(1+x^2)), use the chain rule: (f(g(x)))' = f'(g(x))·g'(x). Example: d/dx[sin(3x)] = cos(3x)·3. 5) For combinations, apply rules in order (expand if helpful), then simplify algebraically. 6) For limits or proving derivatives from first principles, recall key limits like lim_{x→0} (sin x)/x = 1 (LIM-3.A.1). Want worked examples and AP-style practice? Check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv), the Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2), and extra practice problems (https://library.fiveable.me/practice/ap-calculus).

Short answer: the derivative of e^x is exactly e^x. That’s one of the key FUN-3.A facts in Topic 2.7. Why: the natural exponential function e^x is defined so that its rate of change equals its value. Using the limit definition of derivative (LIM-3.A.1) you can show lim_{h→0} (e^{x+h} − e^x)/h = e^x · lim_{h→0} (e^h − 1)/h, and the inner limit equals 1, so d/dx e^x = e^x. One common point of confusion: if the exponent is a function u(x), use the chain rule (FUN-3.A): d/dx e^{u(x)} = u′(x)·e^{u(x)}. So e^{2x} differentiates to 2e^{2x}, not just e^{2x}. If you want a quick AP-aligned refresher, check the Topic 2.7 study guide (https://library.fiveable.me/ap-calculus/unit-2/derivatives-cos-x-sinx-ex-ln-x/study-guide/SbmDK3t2kYI0u2wLz9Hv) and practice more problems at https://library.fiveable.me/practice/ap-calculus.