Use the power rule from the CED: for any constant exponent r, d/dx x^r = r x^(r−1). So for f(x) = x^5, f′(x) = 5x^4. If you want one-line justification: the difference quotient/limit definition yields the same result, but on the exam you can directly apply the power rule for integer (and rational) exponents (FUN-3.A.1). That derivative gives the slope of the tangent line at any x; for example at x = 2, f′(2) = 5·2^4 = 80. For more practice and to see this in AP-style problems, check the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and try problems from the unit practice pool (https://library.fiveable.me/practice/ap-calculus).

The Power Rule: if f(x) = x^r for any real number r (integer, negative, or fractional) and x is in the domain where the expression makes sense, then f '(x) = r·x^(r−1). Examples: - If f(x)=x^5, f '(x)=5x^4. - If f(x)=x^(1/2) (square root), f '(x)=(1/2)x^(−1/2)=1/(2√x). - If f(x)=x^(−2), f '(x)=−2x^(−3). You can combine this with the constant multiple and sum/difference rules to differentiate polynomials term-by-term. For higher-order derivatives, apply the rule repeatedly (e.g., d^2/dx^2 x^n = n(n−1)x^(n−2)). This is exactly the FUN-3.A skill in the AP CED for Topic 2.5. For a quick review, check the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the power rule whenever the piece of the function is a power function x^r (including integer, negative, or rational exponents). For f(x)=x^r, d/dx[x^r]=r x^(r-1)—that’s the fastest route instead of doing the limit definition every time (this is exactly what Topic 2.5/FUN-3.A covers). Switch to other rules when the expression isn’t a simple x^r or is combined with other operations: - Constant multiple, sum/difference rules: use when terms are constants times or sums of simpler pieces (apply power rule to each term). - Product rule: use when two non-constant factors multiply (not separable into a single power). - Quotient rule: use for a fraction of two functions (or rewrite as a power if convenient). - Chain rule: use when you have a composite like (g(x))^r (e.g., (3x+1)^5)—apply power rule to outer function and multiply by derivative of inner function. On the AP exam, identify each term’s form quickly and pick the rule that gives the fewest steps. For more examples and practice aligned to the CED, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk)—Fiveable has lots of practice problems to drill this.

Power Rule (step-by-step)—idea: if f(x) = x^r (r any real number where derivative exists), then f'(x) = r x^(r-1). Step-by-step with an example: 1. Start with f(x) = x^4. We want f'(x) (slope of tangent / instantaneous rate of change). 2. Apply the Power Rule: multiply by the exponent and subtract 1 from the exponent: f'(x) = 4·x^(4−1). 3. Simplify: f'(x) = 4x^3. Why it works (brief): you can get this from the difference quotient and a binomial expansion (or from rules proved in the CED FUN-3.A), but for AP problems you’ll apply the Power Rule directly for integer, negative, or fractional exponents (e.g., d/dx x^(1/2) = (1/2)x^(−1/2)). Use with other rules: combine with constant multiple, sum/difference, product/quotient, and chain rule for composites on the AP exam. For more practice and the Topic 2.5 study guide, see the Applying the Power Rule study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk). For extra problems, try the practice bank (https://library.fiveable.me/practice/ap-calculus).

The power rule works the same for negative exponents: for any real r, d/dx x^r = r x^{r-1}. So for x^(-3): d/dx x^(-3) = -3 x^(-4) = -3 / x^4. Two quick points to avoid mistakes: - You can apply the rule directly on the AP (FUN-3.A in the CED). No need to re-derive from the limit each time. - Remember domain: x^(-3) and its derivative aren’t defined at x = 0, so watch domain restrictions in context problems. If you want a refresher with examples and practice, check the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and the Unit 2 overview (https://library.fiveable.me/ap-calculus/unit-2). For extra practice, use the problems at (https://library.fiveable.me/practice/ap-calculus).

Use the power rule: if f(x) = x^r then f′(x) = r x^(r−1). Here r = 1/2, so f(x) = x^(1/2) ⇒ f′(x) = (1/2) x^(1/2 − 1) = (1/2) x^(−1/2). Rewrite with a root: f′(x) = 1/(2√x). Notes: this applies for x > 0 because √x is only defined (real) for x ≥ 0 and the derivative formula requires x ≠ 0 (x = 0 is not differentiable here). The CED explicitly includes rational exponents in the power-rule scope (FUN-3.A.1), and you could also get the same result from the limit/difference-quotient if needed on an exam. For a short review, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and try practice problems at (https://library.fiveable.me/practice/ap-calculus).

The definition of the derivative is the limit of the difference quotient: f′(x) = lim_{h→0} [f(x+h) − f(x)]/h. You can always use this to find a derivative from first principles—it’s how you justify rules and handle tricky cases. The power rule is a shortcut you use when f(x)=x^r (r an integer, rational, negative, or fractional as allowed by the CED): d/dx[x^r] = r x^{r−1}. Instead of doing the limit every time, you apply this directly to get derivatives fast (this is what FUN-3.A expects you to use). When to use which: - Use the power rule on routine problems to save time on the exam (AP permits direct application; it’s part of Topic 2.5). - Use the limit definition when you need a proof, when a function isn’t obviously a power function, or when endpoint/continuity/domain issues matter. For review and examples, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk). For more practice, try problems at the unit page (https://library.fiveable.me/ap-calculus/unit-2) and the practice bank (https://library.fiveable.me/practice/ap-calculus).

Think of r as a rational number r = p/q (q>0). For x>0 define y = x^{p/q}. Then y^q = x^p. Differentiate both sides implicitly (difference quotient/limit idea under the hood): q y^{q-1} (dy/dx) = p x^{p-1}. Solve for dy/dx and substitute y = x^{p/q} to get dy/dx = (p/q) x^{p/q - 1} = r x^{r-1}. So the same power-rule formula holds for rational (fractional) exponents. Notes: we used implicit differentiation which ultimately rests on the limit definition of derivative, so this aligns with the CED keywords (difference quotient, limit definition). Also watch domain—fractional exponents with even denominators require x>0 unless you use complex numbers. For a clean AP review, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and try practice problems (https://library.fiveable.me/practice/ap-calculus).

Use the power rule just like with integers: if f(x) = x^(r) for any rational r, then f'(x) = r x^(r−1). So for f(x) = x^(2/3), f'(x) = (2/3) x^(−1/3) = 2 / (3 x^(1/3)). Notes you need for the AP: the power rule for rational exponents comes from the limit/definition (FUN-3.A.1), so it’s valid wherever x^(2/3) is defined and differentiable. Here x^(2/3) = (x^(1/3))^2 is defined for all real x, but the derivative blows up at x = 0 because x^(−1/3) → ∞; so f′(0) does not exist (vertical tangent). If you want practice or a quick refresher on applying the power rule to fractional exponents, check the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and try problems at https://library.fiveable.me/practice/ap-calculus.

Yes—√x means x^(1/2). That lets you use the power rule for rational exponents. If f(x) = x^(1/2), then by the power rule (FUN-3.A from the CED): f′(x) = (1/2) x^(1/2 − 1) = (1/2) x^(−1/2) = 1/(2√x). Quick notes you should care about for AP Calc: - This works because the power rule holds for integer, rational, and negative exponents (keywords: power rule, rational exponents). - Domain: √x is defined for x ≥ 0 (real-valued). The derivative 1/(2√x) is only defined for x > 0, and √x is not differentiable at x = 0 (the slope becomes infinite). - You can also derive this from the limit definition if asked on the exam, but on AP problems you’re expected to apply the power rule directly. For a short topic review, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk). For lots of practice, try the AP practice questions (https://library.fiveable.me/practice/ap-calculus).

Short answer: x^0 is the constant function 1 (for x ≠ 0 we have x^0 = 1), so its derivative is 0. Why that matches the Power Rule: the power rule says d/dx[x^r] = r x^{r-1}. Plugging r = 0 gives 0 · x^{-1} = 0 for every x where x^{-1} makes sense (x ≠ 0). More fundamentally, x^0 is constant, so by the constant rule its derivative is 0. Using the limit definition you get the same result: the difference quotient for the constant 1 is 0. Caveat for x = 0: 0^0 is undefined, so if a function is written f(x)=x^0 you should check how it's defined at x = 0. If the function is explicitly defined as f(x)=1 for all x (including 0), then f′(0)=0. If it's left as an expression with 0^0 undefined, you must state the value at 0 before differentiating. For more practice and AP-aligned notes see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and extra problems (https://library.fiveable.me/practice/ap-calculus).

Write 1/x^2 as x^(-2). The power rule says d/dx[x^r] = r x^(r−1) for integer, rational, or negative r (FUN-3.A). So d/dx(1/x^2) = d/dx(x^(−2)) = −2 x^(−3) = −2 / x^3. Domain note: this derivative is valid for x ≠ 0 (both the original function and derivative are undefined at 0). This is a direct, AP-aligned use of the power rule (negative exponent case from Topic 2.5). For a quick refresher on applying the power rule and more examples, check the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and try practice problems (https://library.fiveable.me/practice/ap-calculus) to solidify this.

Yes—the power rule f(x) = x^r => f′(x) = r x^{r−1} works for any real exponent r, not just rational or integer, but with the usual domain caveats. Why: for x>0 you can write x^r = e^{r ln x} and differentiate to get r x^{r−1}, so the formula holds for real r (including π). For x ≤ 0 you must be careful—x^r for irrational r isn’t real-valued when x<0, so the rule doesn’t apply there in the real-number sense. At x = 0 the derivative depends on r: if r>1 the derivative at 0 is 0; if 0https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk). For more practice problems check (https://library.fiveable.me/practice/ap-calculus).

Use the Power Rule (d/dx x^n = n x^(n-1)) plus the constant multiple and sum/difference rules. Step-by-step: 1. f(x) = 3x^4 − 2x^3 + x^2. Treat each term separately. 2. Differentiate 3x^4: pull out the constant 3, then apply the Power Rule: 3 · d/dx(x^4) = 3 · (4x^3) = 12x^3. 3. Differentiate −2x^3: −2 · d/dx(x^3) = −2 · (3x^2) = −6x^2. 4. Differentiate x^2: d/dx(x^2) = 2x. Now combine: fʹ(x) = 12x^3 − 6x^2 + 2x. That’s the derivative you’ll use for tangent slopes, higher-order derivatives, or AP problems asking for polynomial differentiation (FUN-3.A in the CED). For more on the Power Rule and practice, see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and tons of practice questions at (https://library.fiveable.me/practice/ap-calculus).

Short version: the power rule works the same for negative (and fractional) exponents—if f(x) = x^r, then f'(x) = r x^{r-1}, provided x is in the domain where that expression makes sense. Quick steps to avoid mistakes: - Rewrite negative exponents as powers: x^{-3} = 1/x^3. - Apply the power rule directly: d/dx[x^{-3}] = -3 x^{-4} = -3/(x^4). - If you prefer quotient rule, check it gives the same result: d/dx[1/x^3] = 0·x^3 − 1·3x^2 over x^6 = −3/x^4. Example: f(x)=x^{-1/2} → f'(x)=(-1/2)x^{-3/2} = −1/(2 x^{3/2}). Watch domain: x^{-1/2} only for x>0 (real), so answers only valid where the original function is defined. On the AP exam you’ll rely on FUN-3.A (power rule for integer/rational/negative exponents). For a focused review see the Topic 2.5 study guide (https://library.fiveable.me/ap-calculus/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk) and practice hundreds of problems at (https://library.fiveable.me/practice/ap-calculus).