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

More AP Calculus AB guides

• Limits and Continuity
• Applications of Derivatives
• Integration
• Differential Equations

Derivative Definition

f'(a) = lim(h→0) [f(a+h)-f(a)]/h; instantaneous rate of change; slope of tangent line at point (a, f(a)).

Derivative as Limit

Derivative exists when limit exists; geometrically, derivative is slope of tangent; physically, instantaneous velocity.

Tangent Line Equation

At point (a, f(a)), tangent line: y - f(a) = f'(a)(x - a); uses point-slope form with derivative as slope.

Differentiability and Continuity

If f differentiable at a, then f continuous at a. Converse false: continuous doesn't imply differentiable (sharp corners).

Power Rule

d/dx[x^n] = n·x^(n-1); applies to any real n; fundamental rule used constantly in differentiation.

Constant and Constant Multiple Rule

d/dx[c] = 0; d/dx[c·f(x)] = c·f'(x); constants differentiate to 0, constant factors pull out.

Sum and Difference Rule

d/dx[f(x) ± g(x)] = f'(x) ± g'(x); differentiate term-by-term; linearity of derivative.

Product Rule

d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x); 'first times derivative of second plus second times derivative of first.'

Quotient Rule

d/dx[f(x)/g(x)] = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]²; mnemonic: 'low d-high minus high d-low, square the bottom.'

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x); derivative of composition; multiply outer and inner derivatives.

Derivative of Exponential

d/dx[e^x] = e^x; d/dx[a^x] = a^x·ln(a); exponential functions have special property of self-reproduction.

Derivative of Logarithm

d/dx[ln(x)] = 1/x; d/dx[log_a(x)] = 1/(x·ln(a)); inverse relationship with exponential.

Derivative of Trig Functions

d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x), d/dx[tan(x)] = sec²(x); remember signs and patterns.

Inverse Trig Derivatives

d/dx[arcsin(x)] = 1/√(1-x²), d/dx[arctan(x)] = 1/(1+x²); related to derivatives of original trig functions.

Implicit Differentiation

Differentiate both sides with respect to x; treat y as function of x (use chain rule); solve for dy/dx; useful when y can't be isolated.

Related Rates

Two variables related by equation; differentiate with respect to time; find rate of one variable given rate of another.

Logarithmic Differentiation

Take ln of both sides, differentiate, solve for y'; useful for functions with variable exponents or complex products.

Higher Order Derivatives

f''(x) = d/dx[f'(x)]; second derivative; f'''(x) is third derivative; notation: f^(n)(x) for nth derivative.

Concavity and Second Derivative

f''(x) > 0: concave up (U-shaped); f''(x) < 0: concave down (∩-shaped); second derivative test for extrema.

Linear Approximation

Near point a: f(x) ≈ f(a) + f'(a)(x-a); tangent line approximates curve locally; useful for estimating function values.

Logarithmic Differentiation

For y = f(x)^g(x) or messy products, take ln of both sides, differentiate implicitly, then multiply by y to isolate dy/dx.

Derivative of x^x

d/dx[x^x] = x^x(1 + ln x); neither the power rule nor the exponential rule applies alone because base and exponent both vary.

Inverse Function Derivative

If g = f⁻¹ then g'(b) = 1/f'(a) where f(a) = b; the slopes of inverse graphs at corresponding points are reciprocals.

Derivatives of Inverse Trig

d/dx[arcsin u] = u'/√(1-u²), d/dx[arctan u] = u'/(1+u²), d/dx[arcsec u] = u'/(|u|√(u²-1)).

Implicit Differentiation

Differentiate both sides treating y as a function of x, applying the chain rule to every y term, then solve algebraically for dy/dx.

Implicit Second Derivative

After finding dy/dx implicitly, differentiate again and substitute the first-derivative expression wherever dy/dx reappears.

Chain Rule with Tables

For h = f(g(x)), h'(a) = f'(g(a))·g'(a); table problems test whether the inner value is used inside f' before multiplying.

Product of Three Factors

d/dx[uvw] = u'vw + uv'w + uvw'; each factor is differentiated in turn while the others are held.

Quotient Rule Structure

(u/v)' = (u'v - uv')/v²; the order of subtraction matters, and the denominator is squared, not differentiated.

Corner Points

A corner such as |x-3| at x = 3 gives unequal one-sided derivatives, so the function is continuous but not differentiable there.

Vertical Tangents

x^(1/3) at 0 or x^(2/3) at 0 give derivatives blowing up; the tangent is vertical or a cusp forms, so the derivative does not exist.

Differentiability Implies Continuity

Every differentiable point is continuous, but the converse fails; continuity is necessary and not sufficient for a derivative to exist.

Derivative of a^x

d/dx[a^x] = a^x·ln a; the natural exponential is the special case where ln a = 1.

Higher-Order Derivatives of e^(-x)sin x

Repeated differentiation cycles through combinations of sine and cosine; for example the second derivative of e^(-x)sin x is -2e^(-x)cos x.

Normal Line

The line perpendicular to the tangent at a point, with slope -1/f'(a); often requested alongside a tangent-line problem.

Derivative of ln|u|

d/dx[ln|u|] = u'/u; the absolute value extends the formula to negative inputs without changing the derivative.

Velocity from Position

If s(t) gives position, s'(t) is velocity and s''(t) is acceleration; solving s'(t) = 0 finds the instants the object is momentarily at rest.

Symmetry of Derivatives

The derivative of an even function is odd, and the derivative of an odd function is even; a useful check on symbolic work.

Estimating Derivatives from Data

With only tabulated values, approximate f'(a) by a symmetric difference [f(a+h)-f(a-h)]/(2h), which is usually more accurate than a one-sided quotient.

Derivative of a Piecewise Function

Differentiate each branch separately, then check the seam by comparing one-sided derivatives; matching values alone is not enough.