The mathematics of instantaneous change — the most powerful tool for analyzing how things evolve.
The derivative of f at x measures the instantaneous rate of change — the slope of the tangent line. It's defined as a limit:
Geometrically, this is the slope of the tangent line to the graph of f at the point (x, f(x)). The tangent line concept connects to geometric tangency — touching a curve at exactly one point locally.
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These results rely on the limit definitions of sine and the exponential function. See trig identities for deriving the trig derivatives, and exponential functions for the exponential derivative proof.
For composite functions f(g(x)):
"Derivative of the outer, times derivative of the inner." This is the most frequently used rule in all of calculus.
Outer: sin(u) → cos(u). Inner: u = x³ → 3x²
d/dx sin(x³) = cos(x³) · 3x² = 3x² cos(x³)
When y is defined implicitly by an equation (e.g. x² + y² = 25, the equation of a circle), differentiate both sides with respect to x, treating y as a function of x:
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Solve: dy/dx = −x/y
At the point (3, 4): slope = −3/4. This is the slope of the tangent to the circle at that point.
Implicit differentiation is essential for conic sections, for finding slopes of inverse functions (including inverse trig derivatives), and in differential equations.
Find maximum and minimum values by setting f'(x) = 0 and analyzing using the second derivative test.
Constraint: 2l + 2w = 100 → w = 50 − l
Area: A = l(50 − l) = 50l − l²
A'(l) = 50 − 2l = 0 → l = 25, w = 25
Maximum area = 625 (a square, as geometry might suggest!)
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V = (4/3)πr³ → dV/dt = 4πr² · (dr/dt) = 4π(100)(2) = 800π cm³/s
This is the tangent line approximation — the simplest case of Taylor series.