Derivatives
Differentiate with the limit definition and the power rule, evaluate f′(a), and write tangent lines — with every step shown.
1 The derivative as a limit
The derivative is the limit of the slopes of secant lines as the second point slides into the first.
The fraction is the slope between (x, f(x)) and (x + h, f(x + h)). Letting h → 0 squeezes the secant into the tangent — so f′(x) is the slope of the tangent line at x, i.e. the instantaneous rate of change.
[(x+h)² − x²]/h = [2xh + h²]/h = 2x + h → 2x as h → 0.
2 The power rule
The limit definition always works — but the power rule does it in one step.
d/dx [a·xn] = a·n·xn−1 (constant multiple)
d/dx [f + g] = f′ + g′ (sum rule)
Apply term by term: d/dx [3x² + 2x] = 6x + 2. Constants vanish: d/dx [7] = 0.
12x² − 1 = 12x² − 1.
3 Tangent lines
The derivative at a point is the slope of the tangent line there.
2. point: (a, f(a))
3. line: y − f(a) = m(x − a)
Never confuse f(a) (the height) with f′(a) (the slope): the point uses f, the slope uses f′.
f′(x) = 2x, so m = 6; f(3) = 9. Line: y − 9 = 6(x − 3), i.e. y = 6x − 9.
4 Worked examples
Each claim verified independently.
Find d/dx [x5].
5x4: 5x^4.
Find d/dx [x].
1.
Find d/dx [3x2 + 2x].
6x + 2: 6x + 2.
f(x) = 2x2. Find f′(3).
f′(x) = 4x; f′(3) = 12.
For f(x) = x2, simplify [f(x + h) − f(x)]/h.
2x + h (then h → 0 gives 2x).
5 Common mistakes
Three errors that break derivative work.
Wrong: d/dx [x5] = 5x5.
Right: the power rule does two things — multiply by n and subtract 1 from the exponent: 5x4.
Wrong: tangent to x² at x = 3 has slope f(3) = 9.
Right: slope comes from the derivative: f′(3) = 6. The point (3, 9) uses f; the slope uses f′.
Wrong: from the definition, f′(x) = 2x + h.
Right: 2x + h is the difference quotient; the derivative takes h → 0, giving 2x.
6 Key vocabulary
Words to know
- Difference quotient — [f(x + h) − f(x)]/h, the secant slope.
- Derivative — f′(x): the limit of the difference quotient; tangent slope.
- Tangent line — y − f(a) = f′(a)(x − a).
- Power rule — d/dx [xn] = n·xn−1.
7 Quick check
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8 Next steps
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