Derivative Machine: Power and Chain Rules

MTH-117 · Semester I · Mathematics I

beginner20 minAvailable

Limits of quotients

Guided practice — solve each problem, check, then generate a fresh one.

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Formula to apply

lim_{x→a} (x² − a²)/(x − a) = 2a

Problem

Evaluate lim_{x→2} (x² − 4) / (x − 2).

limit

Working

Lab brief

Fluently apply the power rule, sum rule and chain rule to polynomials and exponentials, evaluating derivatives at a point.

Theory & concepts

Derivative

f′(x) = lim[h→0] (f(x + h) − f(x))/h. It is the instantaneous rate of change — the slope of the tangent line.

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Rules for finding derivatives

The power rule says d/dx[xⁿ] = n·xⁿ⁻¹. Constants drop out, and the derivative of a sum is the sum of the derivatives.

The chain rule handles nested functions: d/dx[e^{kx}] = k·e^{kx}. The exponential is the only function equal to its own derivative up to this factor.

Common slip

d/dx[xⁿ] is n·xⁿ⁻¹, not n·xⁿ. Check that the exponent has dropped by exactly one before you move on.

f(x) = 2x³ − x² + 4x − 1, find f′(1)
f′(x) = 6x² − 2x + 4 → f′(1) = 8.
f(x) = 3e^{2x}, find f′(0)
f′(x) = 6e^{2x} → f′(0) = 6.
Interpret the number
Explain in one sentence what the derivative you computed means about the graph at that x.

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