A complete drill-and-discover resource covering the Power Rule for differentiation — from first principles through polynomial, radical, and rational function applications. Suitable for AP Calculus AB/BC, university Calculus I, and advanced high-school mathematics.
Let \(n\) be any real number and \(f(x)=x^{n}\). Then:
Valid for all real \(n\); note \(x\neq0\) when \(n<1\).
| Function | Rewrite as Power | Derivative |
|---|---|---|
| \(\sqrt{x}\) | \(x^{1/2}\) | \(\dfrac{1}{2}x^{-1/2}=\dfrac{1}{2\sqrt{x}}\) |
| \(\sqrt[3]{x}\) | \(x^{1/3}\) | \(\dfrac{1}{3}x^{-2/3}\) |
| \(\sqrt[n]{x^{m}}\) | \(x^{m/n}\) | \(\dfrac{m}{n}x^{(m/n)-1}\) |
| \(\dfrac{1}{x}\) | \(x^{-1}\) | \(-x^{-2}=-\dfrac{1}{x^{2}}\) |
| \(\dfrac{1}{x^{n}}\) | \(x^{-n}\) | \(-n\,x^{-n-1}\) |
| \(c\) (constant) | \(c\cdot x^{0}\) | \(0\) |
Example 1. Find \(\dfrac{d}{dx}\!\left[x^{7}\right]\).
With \(n=7\): \(\dfrac{d}{dx}\!\left[x^{7}\right]=7x^{6}\).
Example 2. Differentiate \(f(x)=3x^{4}-5x^{2}+7\).
Term-by-term: \(f'(x)=12x^{3}-10x+0=12x^{3}-10x\).
Example 3. Differentiate \(g(x)=\dfrac{4}{x^{3}}\).
Rewrite: \(g(x)=4x^{-3}\). Then \(g'(x)=4(-3)x^{-4}=-\dfrac{12}{x^{4}}\).
Example 4. Differentiate \(h(x)=5\sqrt[4]{x^{3}}\).
Rewrite: \(h(x)=5x^{3/4}\). Then \(h'(x)=5\cdot\dfrac{3}{4}x^{-1/4}=\dfrac{15}{4}x^{-1/4}\).
Example 5. Differentiate \(f(x)=\dfrac{x^{3}-2x+1}{x}\).
Simplify first: \(f(x)=x^{2}-2+x^{-1}\). Then \(f'(x)=2x-x^{-2}=2x-\dfrac{1}{x^{2}}\).
Differentiate each function. Express answers using positive exponents unless otherwise stated.
Rewrite radicals and rational expressions as powers first, then differentiate.
Multi-step reasoning, tangent lines, higher-order derivatives, and real-world interpretation.
Circle the best answer. These mirror AP Calculus and university exam formats.
\(f(x)=x^{7},\quad n=7\)
Power Rule: multiply by the exponent, then reduce the exponent by 1.
\(f(x)=x^{10},\quad n=10\)
\(g(x)=x^{-3},\quad n=-3\)
\(g'(x)=(-3)x^{-3-1}=-3x^{-4}\)
\(h(x)=x^{1/2},\quad n=\tfrac{1}{2}\)
\(h'(x)=\tfrac{1}{2}x^{1/2-1}=\tfrac{1}{2}x^{-1/2}\)
\(f(x)=x^{2/3},\quad n=\tfrac{2}{3}\)
\(f'(x)=\tfrac{2}{3}x^{2/3-1}=\tfrac{2}{3}x^{-1/3}\)
\(f(x)=x^{-1/4},\quad n=-\tfrac{1}{4}\)
\(f'(x)=-\tfrac{1}{4}x^{-1/4-1}=-\tfrac{1}{4}x^{-5/4}\)
\(g(x)=5x^{4}\). Apply Constant Multiple Rule + Power Rule:
\(h(x)=-3x^{6}\)
\(f(x)=\tfrac{1}{2}x^{8}\)
\(g(x)=\pi x^{3}\). Here \(\pi\) is a constant coefficient.
\(f(x)=x^{0}=1\) for \(x\neq0\) — a constant function.
\(h(x)=7\) — constant function. Constant Rule applies directly.
\(f(x)=x^{100},\quad n=100\)
\(g(x)=x^{-5},\quad n=-5\)
\(g'(x)=-5x^{-5-1}=-5x^{-6}\)
\(f(x)=x^{3/4},\quad n=\tfrac{3}{4}\)
\(f'(x)=\tfrac{3}{4}x^{3/4-1}=\tfrac{3}{4}x^{-1/4}\)
\(h(x)=x^{5/2},\quad n=\tfrac{5}{2}\)
\(h'(x)=\tfrac{5}{2}x^{5/2-1}=\tfrac{5}{2}x^{3/2}\)
\(f(x)=x^{-2/3},\quad n=-\tfrac{2}{3}\)
\(f'(x)=-\tfrac{2}{3}x^{-2/3-1}=-\tfrac{2}{3}x^{-5/3}\)
\(g(x)=4x^{-7}\)
\(g'(x)=4\cdot(-7)x^{-8}=-28x^{-8}\)
\(f(x)=-\tfrac{2}{3}x^{9}\)
\(f'(x)=-\tfrac{2}{3}\cdot9x^{8}=-6x^{8}\)
Rewrite: \(h(x)=\sqrt{x^{5}}=(x^{5})^{1/2}=x^{5/2}\)
\(h'(x)=\tfrac{5}{2}x^{5/2-1}=\tfrac{5}{2}x^{3/2}\)
\(f(x)=3x^{4}-5x^{2}+7\). Differentiate term-by-term:
\(\dfrac{d}{dx}[3x^{4}]=12x^{3},\quad \dfrac{d}{dx}[-5x^{2}]=-10x,\quad \dfrac{d}{dx}[7]=0\)
\(g(x)=x^{3}-4x+1\)
\(h(x)=8x^{5}-3x^{3}+2x-11\)
Rewrite: \(f(x)=\dfrac{1}{x^{4}}=x^{-4}\)
\(f'(x)=(-4)x^{-5}\)
Rewrite: \(g(x)=3x^{-2}\)
\(g'(x)=3\cdot(-2)x^{-3}=-6x^{-3}\)
Rewrite: \(h(x)=5x^{-6}-2x^{-1}\)
\(h'(x)=5(-6)x^{-7}-2(-1)x^{-2}=-30x^{-7}+2x^{-2}\)
Rewrite: \(f(x)=x^{1/2}+x^{1}\)
\(f'(x)=\tfrac{1}{2}x^{-1/2}+1\)
Rewrite: \(g(x)=3x^{1/2}-2x^{-1/2}\)
\(g'(x)=3\cdot\tfrac{1}{2}x^{-1/2}-2\cdot\!\left(-\tfrac{1}{2}\right)x^{-3/2}=\tfrac{3}{2}x^{-1/2}+x^{-3/2}\)
Rewrite: \(h(x)=\sqrt[3]{x^{4}}=x^{4/3}\)
\(h'(x)=\tfrac{4}{3}x^{4/3-1}=\tfrac{4}{3}x^{1/3}\)
Rewrite: \(f(x)=x^{2}+x^{1/4}\)
\(f'(x)=2x+\tfrac{1}{4}x^{1/4-1}=2x+\tfrac{1}{4}x^{-3/4}\)
Simplify: \(g(x)=\dfrac{x^{3}+2x}{x}=x^{2}+2\)
Simplify: \(h(x)=\dfrac{x^{4}-x^{2}+1}{x^{2}}=x^{2}-1+x^{-2}\)
\(h'(x)=2x+0+(-2)x^{-3}=2x-2x^{-3}\)
Expand: \(\left(x^{2}+1\right)^{2}=x^{4}+2x^{2}+1\)
Expand: \(\left(2x-3\right)^{2}=4x^{2}-12x+9\)
\(h(x)=x^{-1/2}+x^{-3/2}\)
\(h'(x)=-\tfrac{1}{2}x^{-3/2}+\!\left(-\tfrac{3}{2}\right)x^{-5/2}\)
\(f(x)=6x^{2/3}-9x^{1/3}\)
\(f'(x)=6\cdot\tfrac{2}{3}x^{-1/3}-9\cdot\tfrac{1}{3}x^{-2/3}=4x^{-1/3}-3x^{-2/3}\)
Simplify: \(g(x)=\dfrac{4x^{3}-6x^{2}}{2x}=2x^{2}-3x\)
\(h(t)=t^{4}-2t^{3}+t-5\)
\(f(s)=s^{3/2}-4s^{1/2}+2\)
\(f'(s)=\tfrac{3}{2}s^{1/2}-4\cdot\tfrac{1}{2}s^{-1/2}=\tfrac{3}{2}s^{1/2}-2s^{-1/2}\)
\(g(u)=u^{-4}-u^{-2}+u\)
\(g'(u)=(-4)u^{-5}-(-2)u^{-3}+1=-4u^{-5}+2u^{-3}+1\)
Simplify: \(h(x)=\dfrac{\sqrt{x}+1}{\sqrt{x}}=1+x^{-1/2}\)
\(h'(x)=0+(-\tfrac{1}{2})x^{-3/2}\)
Simplify: \(f(x)=\dfrac{2x^{4}-3x^{3}+x}{x^{2}}=2x^{2}-3x+x^{-1}\)
\(f'(x)=4x-3+(-1)x^{-2}=4x-3-x^{-2}\)
\(g(x)=x^{5/3}+x^{-5/3}\)
\(g'(x)=\tfrac{5}{3}x^{5/3-1}+(-\tfrac{5}{3})x^{-5/3-1}=\tfrac{5}{3}x^{2/3}-\tfrac{5}{3}x^{-8/3}\)
Expand: \(\left(x^{1/2}+x^{-1/2}\right)^{2}=x+2\cdot x^{1/2}\cdot x^{-1/2}+x^{-1}=x+2+x^{-1}\)
\(h'(x)=1+0+(-1)x^{-2}=1-x^{-2}\)
\(f(x)=3x^{-2}-5x^{-4}+x^{-6}\)
\(f'(x)=3(-2)x^{-3}-5(-4)x^{-5}+(-6)x^{-7}=-6x^{-3}+20x^{-5}-6x^{-7}\)
\(f(x)=x^{3}-2x\implies f'(x)=3x^{2}-2\)
At \(x=1\): slope \(=f'(1)=3(1)^{2}-2=1\). Point: \(f(1)=1-2=-1\), so \((1,-1)\).
Line: \(y-(-1)=1\cdot(x-1)\)
\(g(x)=x^{1/2}\implies g'(x)=\tfrac{1}{2}x^{-1/2}\)
At \(x=4\): slope \(=\tfrac{1}{2}(4)^{-1/2}=\tfrac{1}{4}\). Point: \(g(4)=2\), so \((4,2)\).
Line: \(y-2=\tfrac{1}{4}(x-4)\)
\(y=x^{3}-3x\implies y'=3x^{2}-3\)
Set \(y'=0\): \(3x^{2}-3=0\implies x^{2}=1\implies x=\pm1\)
\(y(1)=1-3=-2\), \(y(-1)=-1+3=2\)
\(y=x^{4}-8x^{2}+3\implies y'=4x^{3}-16x=4x(x^{2}-4)=4x(x-2)(x+2)\)
Roots: \(x=0,\;x=2,\;x=-2\)
\(y(0)=3,\quad y(2)=16-32+3=-13,\quad y(-2)=-13\)
\(f(x)=x^{3}-12x+5\implies f'(x)=3x^{2}-12\)
Set \(f'(x)=0\): \(3x^{2}=12\implies x^{2}=4\)
\(s(t)=t^{3}-6t^{2}+9t\implies v(t)=s'(t)=3t^{2}-12t+9\)
At \(t=2\): \(v(2)=3(4)-12(2)+9=12-24+9=-3\)
Set \(v(t)=3t^{2}-12t+9=0\implies3(t-1)(t-3)=0\)
\(h(t)=-16t^{2}+64t+80\implies v(t)=h'(t)=-32t+64\)
At \(t=1\): \(v(1)=-32+64=32\)
Maximum height when \(v(t)=0\): \(-32t+64=0\implies t=2\text{ s}\).
\(h(2)=-16(4)+64(2)+80=-64+128+80=144\text{ ft}\)
\(R(x)=50x-0.5x^{2}\implies R'(x)=50-x\)
At \(x=40\): \(R'(40)=50-40=10\)
\(C(x)=0.01x^{3}-0.6x^{2}+13x+100\implies C'(x)=0.03x^{2}-1.2x+13\)
At \(x=20\): \(C'(20)=0.03(400)-1.2(20)+13=12-24+13=1\)
\(f(x)=cx^{4}\implies f'(x)=4cx^{3}\)
Given \(f'(2)=48\): \(4c(2)^{3}=48\implies4c\cdot8=48\implies32c=48\implies c=\tfrac{48}{32}\)
\(f(x)=ax^{b}\implies f'(x)=abx^{b-1}\)
Condition 1: \(f(1)=a\cdot1^{b}=a=3\)
Condition 2: \(f'(1)=ab\cdot1^{b-1}=ab=3b=12\implies b=4\)
\(f(x)=x^{2/3}\implies f'(x)=\tfrac{2}{3}x^{-1/3}=\dfrac{2}{3x^{1/3}}\)
As \(x\to0\), \(|f'(x)|\to\infty\), so the derivative is undefined at \(x=0\).
Geometrically: the graph has a cusp (sharp point with a vertical tangent) at the origin.
\(f(x)=5x^{4}-3x^{2}+7x-1\)
\(f'(x)=20x^{3}-6x+7\)
\(g(x)=x^{5}-4x^{3}\)
\(g'(x)=5x^{4}-12x^{2}\)
\(g''(x)=20x^{3}-24x\)
\(y=ax^{2}+bx+3\implies y'=2ax+b\)
Through \((1,5)\): \(a+b+3=5\implies a+b=2\quad\cdots(1)\)
Slope 4 at \(x=1\): \(2a+b=4\quad\cdots(2)\)
Subtract \((1)\) from \((2)\): \(a=2\). Then \(b=0\).
\(A=\pi r^{2}\implies\dfrac{dA}{dr}=2\pi r\)
Geometric meaning: \(2\pi r\) is the circumference. Adding a thin ring of width \(dr\) increases area by circumference \(\times dr\).
\(V=\dfrac{4}{3}\pi r^{3}\implies\dfrac{dV}{dr}=4\pi r^{2}\)
Geometric meaning: \(4\pi r^{2}\) is the surface area. Expanding radius by \(dr\) adds volume \(\approx\) surface area \(\times dr\).
Method 1 (Expand): \((x-1)^{4}=x^{4}-4x^{3}+6x^{2}-4x+1\)
\(f'(x)=4x^{3}-12x^{2}+12x-4\)
Method 2 (Pattern): \(\dfrac{d}{dx}[(x-1)^{4}]=4(x-1)^{3}\cdot1=4(x-1)^{3}\)
Expand: \(4(x^{3}-3x^{2}+3x-1)=4x^{3}-12x^{2}+12x-4\quad\checkmark\)
\(f(x)=x^{-3}\implies f'(x)=(-3)x^{-4}=-3x^{-4}\)
\(y=x^{3}\implies y'=3x^{2}\). At \(x=-2\): \(3(-2)^{2}=3\cdot4=12\).
Rewrite: \(g(t)=t^{2/3}\implies g'(t)=\tfrac{2}{3}t^{-1/3}\).
This equals \(\tfrac{2}{3}t^{-1/3}\) (choice B) and \(\tfrac{2}{3\,t^{1/3}}\) (choice C) — both are identical expressions.
\(f(x)=x^{1/3}\implies f'(x)=\tfrac{1}{3}x^{-2/3}=\dfrac{1}{3x^{2/3}}\)
As \(x\to0\), \(f'(x)\to\infty\): vertical tangent, not differentiable at \(x=0\).
\(f(x)=cx^{n}\implies f'(x)=cnx^{n-1}=10x^{4}\implies n-1=4\implies n=5\) and \(cn=10\implies c=2\).
\(f(x)=x^{4}-6x^{2}+5\implies f'(x)=4x^{3}-12x\implies f''(x)=12x^{2}-12\)
At \(x=1\): \(f''(1)=12-12=0\)
\(s(t)=2t^{3}-9t^{2}+12t\implies v(t)=6t^{2}-18t+12\implies a(t)=12t-18\)
At \(t=1\): \(a(1)=12-18=-6\)
Antidifferentiate \(6x^{2}-4x+1\): the general antiderivative is \(2x^{3}-2x^{2}+x+C\).
Both (A) (\(C=0\)) and (C) (\(C=7\)) satisfy this — infinitely many functions share the same derivative.
Rewrite: \(f(x)=3x^{-2}-2x^{-3}\)
\(f'(x)=3(-2)x^{-3}-2(-3)x^{-4}=-6x^{-3}+6x^{-4}\)
This equals \(-\dfrac{6}{x^{3}}+\dfrac{6}{x^{4}}\), so (A) and (C) are equivalent.
The Power Rule needs a constant exponent and variable base.
\(e^{x}\) and \(3^{x}\): variable exponents — need the exponential rule.
\(\ln(x^{3})\): logarithmic function — needs the log rule.
\(x^{\sqrt{2}}\): exponent \(\sqrt{2}\) is a constant, so the Power Rule applies directly.
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