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Unit 2
5–10% of examThe derivative tells you how fast a function is changing at a single point, which is the slope of its tangent line. You'll define it with a limit, estimate it from graphs and tables, see where it fails to exist, and then learn the shortcut rules that make finding derivatives quick.
Longer videos that cover the whole unit. Good for a first pass or a final review.
The average rate of change on [a, b] is (f(b) − f(a)) / (b − a), the slope of a secant line. Shrinking the interval toward one point turns that average into the instantaneous rate of change, the slope of the tangent line.
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The derivative at x = a is the limit of the difference quotient: f′(a) = lim (h→0) (f(a + h) − f(a)) / h, or the equivalent form lim (x→a) (f(x) − f(a)) / (x − a). You'll see it written as f′(x), y′ or dy/dx.
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When all you have is a table or a graph, estimate the derivative at a point with the slope between nearby data points, ideally ones on either side of it. A graphing calculator can also find a numerical estimate.
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Having a derivative at a point forces a function to be continuous there, but the reverse fails: a continuous function can still have no derivative at a point. Corners, cusps and vertical tangent lines are the classic examples, and any discontinuity also rules out a derivative.
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The power rule says the derivative of xⁿ is n·xⁿ⁻¹. Rewrite roots and fractions as powers first, such as √x as x to the ½ power and 1/x² as x⁻², so the rule applies.
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The derivative of a constant is 0, and you can differentiate sums and differences one term at a time. A constant multiplier just comes along: the derivative of k·f(x) is k·f′(x).
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Four derivatives to memorize: sin x → cos x, cos x → −sin x, eˣ → eˣ, and ln x → 1/x (for x > 0). eˣ is special because it is its own derivative.
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The derivative of f(x)·g(x) is f′(x)g(x) + f(x)g′(x). It is not the product of the two derivatives, which is a common mistake.
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The derivative of f(x)/g(x) is (f′(x)g(x) − f(x)g′(x)) / (g(x))². Because of the subtraction, the order of the terms on top matters.
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Writing tan, cot, sec and csc in terms of sin and cos and using the quotient rule gives their derivatives: tan x → sec²x, cot x → −csc²x, sec x → sec x·tan x, and csc x → −csc x·cot x.
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