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Derivatives

The mathematics of instantaneous change — the most powerful tool for analyzing how things evolve.

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The Derivative as a Limit

The derivative of f at x measures the instantaneous rate of change — the slope of the tangent line. It's defined as a limit:

f'(x) = lim (h→0) [f(x + h) − f(x)] / h

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.

Example: Find f'(x) from the definition for f(x) = x²

f'(x) = lim (h→0) [(x+h)² − x²]/h = lim (h→0) [2xh + h²]/h = lim (h→0) (2x + h) = 2x

Differentiation Rules

Power Rule: d/dx [xⁿ] = nxⁿ⁻¹
Constant Multiple: d/dx [c·f(x)] = c·f'(x)
Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x)
Product Rule: d/dx [f·g] = f'g + fg'
Quotient Rule: d/dx [f/g] = (f'g − fg') / g²

Derivatives of Key Functions

d/dx [eˣ] = eˣ     d/dx [ln x] = 1/x
d/dx [sin x] = cos x     d/dx [cos x] = −sin x
d/dx [tan x] = sec²x     d/dx [aˣ] = aˣ · ln(a)

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.

The Chain Rule

For composite functions f(g(x)):

d/dx [f(g(x))] = f'(g(x)) · g'(x)

"Derivative of the outer, times derivative of the inner." This is the most frequently used rule in all of calculus.

Example: Differentiate sin(x³)

Outer: sin(u) → cos(u). Inner: u = x³ → 3x²

d/dx sin(x³) = cos(x³) · 3x² = 3x² cos(x³)

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Implicit Differentiation

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:

Example: Find dy/dx for x² + y² = 25

Differentiate: 2x + 2y(dy/dx) = 0

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.

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Implicit differentiation is essential for conic sections, for finding slopes of inverse functions (including inverse trig derivatives), and in differential equations.

Applications of Derivatives

Optimization

Find maximum and minimum values by setting f'(x) = 0 and analyzing using the second derivative test.

Example: Maximize the area of a rectangle with perimeter 100

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!)

Related Rates

If two quantities are related by an equation, their rates of change are related via the chain rule:

Example: A balloon's radius grows at 2 cm/s. How fast does volume grow when r = 10?

V = (4/3)πr³ → dV/dt = 4πr² · (dr/dt) = 4π(100)(2) = 800π cm³/s

Linear Approximation

f(x) ≈ f(a) + f'(a)(x − a)    (near x = a)

This is the tangent line approximation — the simplest case of Taylor series.

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Derivatives connect to every corner of mathematics. The eigenvalues of the Hessian matrix (matrix of second derivatives) determine whether a multivariable critical point is a max, min, or saddle point. Differential equations are equations written in terms of derivatives. In statistics, maximum likelihood estimation requires setting derivatives equal to zero.

Related Topics

Limits & Continuity The derivative IS a limit — limits are the prerequisite. Integrals Integration is the reverse of differentiation — connected by the FTC. First-Order DEs Equations involving derivatives — the next step after mastering differentiation. Trig Identities Essential for differentiating and simplifying trig expressions.