Derivatives: every key term you need (+ practice quiz)

More AP Calculus BC guides

• Limits & Continuity
• Applications of Derivatives
• Integration
• Applications of Integrals
• Parametric & Polar

Derivative Definition

f'(x) = lim(h→0) [f(x+h) - f(x)] / h. Rate of change of f at x. Slope of tangent line.

Power Rule

d/dx[x^n] = n·x^(n-1). Most fundamental rule. Example: d/dx[x³] = 3x².

Product Rule

(fg)' = f'g + fg'. Derivative of product is DIFFERENT from product of derivatives.

Quotient Rule

(f/g)' = (f'g - fg') / g². Low·d-high minus high·d-low, over low squared.

Chain Rule

(f∘g)' = f'(g(x))·g'(x). Derivative of composition. Critical for complex functions.

Exponential Rule

d/dx[e^x] = e^x. d/dx[a^x] = a^x·ln(a).

Logarithmic Rule

d/dx[ln(x)] = 1/x. d/dx[log_a(x)] = 1/(x·ln(a)).

Trigonometric Derivatives

d/dx[sin(x)]=cos(x), d/dx[cos(x)]=-sin(x), d/dx[tan(x)]=sec²(x).

Inverse Trig Derivatives

d/dx[arcsin(x)] = 1/√(1-x²), d/dx[arctan(x)] = 1/(1+x²).

Implicit Differentiation

Differentiate both sides of equation; solve for dy/dx. Used when can't solve for y explicitly.

Related Rates

Two quantities related by equation; differentiate with respect to time. Example: ladder sliding down wall.

Unit 2 Summary

Derivative formulas: power, product, quotient, chain rules. Applies to exponential, logarithmic, trig functions. Implicit differentiation handles complex relationships.

Differentiability Implies Continuity

If f'(a) exists then f is continuous at a. The converse fails: |x| is continuous but not differentiable at 0.

Corner, Cusp, Vertical Tangent

Three ways differentiability fails while continuity holds: unequal finite one-sided slopes (corner), opposite infinite slopes (cusp), and a common infinite slope (vertical tangent).

Symmetric Difference Quotient

[f(a+h)-f(a-h)]/(2h) estimates f'(a) and is usually more accurate than a one-sided quotient, but it can return a value even where f'(a) fails to exist.

Product Rule for Three Factors

(fgh)' = f'gh + fg'h + fgh'. Each term differentiates exactly one factor and leaves the others intact.

Quotient Rule Memory

(f/g)' = (g f' - f g')/g². The order matters: numerator derivative first, and the whole thing is divided by the square of the denominator.

Chain Rule Layering

For f(g(h(x))), the derivative is f'(g(h(x)))·g'(h(x))·h'(x). Peel one layer at a time from the outside in.

Implicit Differentiation

Differentiate both sides in x, applying the chain rule to every y term to produce dy/dx, then solve algebraically for dy/dx.

Derivative of an Inverse

(f⁻¹)'(b) = 1/f'(a) where f(a)=b. Locate the matching input first; the reciprocal is evaluated at that point, not at b.

Logarithmic Differentiation

Take ln of both sides before differentiating when the expression is a large product, quotient, or has a variable base and variable exponent.

Derivatives of Inverse Trig

d/dx arcsin x = 1/√(1-x²); arccos x is its negative; arctan x = 1/(1+x²); arcsec x = 1/(|x|√(x²-1)).

Hyperbolic-Style Exponential Rules

d/dx a^x = a^x ln a and d/dx log_a x = 1/(x ln a). The natural base is the special case where ln a = 1.

Higher-Order Notation

f''(x) = d²y/dx² measures the rate of change of the slope. In implicit problems you must re-apply the chain rule to any surviving dy/dx.

Second Derivative of a Parametric Curve

d²y/dx² = (d/dt[dy/dx])/(dx/dt). Dividing by dx/dt, not by dt, is the step students most often skip.

Differentiability of Piecewise Functions

Matching values makes a piecewise function continuous; matching one-sided derivatives at the same point makes it differentiable. Both conditions are needed.

Estimating Derivatives from Data

Given a table, approximate f'(c) with the difference quotient over the smallest interval containing c; state whether the estimate is one-sided.

Derivative as a Rate

Units of f'(x) are units of f divided by units of x. Reading units correctly is often the fastest check on an applied answer.

Tangent Line Approximation

L(x) = f(a) + f'(a)(x-a) approximates f near a. The approximation overestimates where f is concave down and underestimates where f is concave up.

Even and Odd Derivatives

The derivative of an even function is odd, and the derivative of an odd function is even. Symmetry flips each time you differentiate.

Local Linearity

Zooming in far enough on a differentiable graph makes it indistinguishable from its tangent line; nondifferentiable points never flatten out this way.

Chain Rule with Tables

If h(x)=f(g(x)), then h'(2)=f'(g(2))·g'(2). Look up the inner output first, then read f' at that value rather than at 2.