In This Lesson The Derivative as a Limit Differentiation Rules The Chain Rule Implicit Differentiation 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 Applications of Derivatives The Derivative as a Limit 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 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
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Derivatives of Key Functions S8sWZpPjMllfNuNwEizobbj11+7PD07nux7wLF5I/gKoGARadFmK22/xA7mJX6n8Mcl6y79ZswYyziJYpO/33rA+EJDpDs/jW3znMI3RA96fNXlWEUQeG+TilU/5xkYTikCvHpbFrxSTsJzXXGFXk7Jq4q5v2+eqPi4aKPIJ4nW5qBTDRdNNFiKN7QBDVBrJ+p/OAgf2059dZKbVwjxm4dnqQGWNaLne2OchU2SPveveUEWGQteupSeX6ZZrG0pdhbEaVL5Ulq5/8QO5iVPjyhXwXC5u707nvS//l9+bNBGx6TaWQcZ5TJmf5+Zebm55ywnFhN7ZBk6qTa6GdscmXOmoF0c8+umRtJu41Rubjc6LKfVPsv8WjvTErthN61fTU7EdW9xjRdM3x412ip/XSx4yaNTyzlwikxI5FDPuk6BsU4sEcAT43cqMOuaJLrYXEa4y7uA93pKrXVJWTH/xn/75LoJluGWUOg8XdtjTprLRNJHEtR7d0gKTW/An01DIAM1ZxlLHtfCQTJRKmmRXIlRKzgL1jO28WIerIbWmeXMVucnxNkiI6FZcqrlrMLxzl/f5jneFT9gpABkTPP/2H12PIxEeI42GRA4f+LK5FGIKX3WKaOC2oU7+/zQQugXhUUkb5Uzv3ficMW+P75a0e4FmPj50mT5H+WlMwp1MpxlOXM0Xy/m37kLDWl+R28oLHU734dGWUpjdolLAPJ0G99lVcaNXOBDV0ItJ2jQN65RSjgiyvR33zOgzVDmpNIPe2/c/jIZT36/sYG+CnoFvI2PkgF2wVwa4pF0A1WYQFApxwB2P8aIfuu+P1X6JolbML4BAyIvP96etCI6pKa+3kMpevR8GoP
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)
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:
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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.
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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
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Maximum area = 625 (a square, as geometry might suggest!)
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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 .
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.