Derivatives & differentiation rules
The idea
The derivative f'(x) is the limit of the difference quotient (f(x + h) − f(x))/h as h → 0: the exact rate of change of f at a single point, and the slope of the tangent line there. You already understand slope as rise over run for lines; differentiation extends that idea to curves by zooming in until the curve is indistinguishable from a line. At university this is the workhorse concept — every model with a changing quantity speaks in derivatives.
Computing from the limit definition every time would be painful, so you build a toolkit: the power rule (d/dx of x^n is n x^(n−1)), linearity, the product rule (uv)' = u'v + uv', the quotient rule, and above all the chain rule, which says the derivative of f(g(x)) is f'(g(x)) × g'(x) — multiply the outer rate by the inner rate. The right mental model is decomposition: read an expression as a recipe of nested operations, then differentiate layer by layer from the outside in.
A persistent misconception is that the product rule should be (uv)' = u'v', differentiating each factor. A quick counterexample kills it: with u = v = x, that would give (x²)' = 1, but the power rule says 2x. Rates of products mix the pieces — each factor takes a turn changing while the other holds still.
Worked example
Let f(x) = x² e^(3x). Find f'(x), then compute the slope of the tangent line at x = 1.
- Recognize the structure: f is a product of u(x) = x² and v(x) = e^(3x), so the product rule applies, and v itself is a composition needing the chain rule.
- Differentiate each factor: u'(x) = 2x by the power rule, and v'(x) = 3 e^(3x) because the chain rule multiplies e^(3x) by the derivative of the inner function 3x.
- Assemble with the product rule: f'(x) = u'v + uv' = 2x e^(3x) + x² × 3 e^(3x) = e^(3x)(2x + 3x²).
- Evaluate at x = 1: f'(1) = e³(2 + 3) = 5e³ ≈ 5 × 20.086 ≈ 100.4, so the tangent there is very steep.
- Sanity-check the form: both terms of f'(x) keep the factor e^(3x), which never vanishes, so f'(x) = 0 only where 2x + 3x² = x(2 + 3x) = 0, at x = 0 and x = −2/3 — sensible critical points for this shape.
Answer. f'(x) = e^(3x)(2x + 3x²), and the tangent slope at x = 1 is 5e³ ≈ 100.4.
Check your understanding
- Why does the chain rule multiply two rates together — what is each rate measuring in the composition?
- How would you convince someone the product rule cannot simply be the product of the derivatives?
- What does the difference quotient (f(x + h) − f(x))/h represent geometrically before the limit is taken?
- Which differentiation rule would you reach for first on x² sin(x⁵), and how do you see that structure quickly?
Build the foundations first
Derivatives & differentiation rules builds on these concepts. If any feel shaky, start there.