AP Calculus BC Unit 2 Notes: Basic Differentiation Rules (Definition to Toolkit)

Derivative

A measure of how fast a function’s output changes as its input changes; the slope of the tangent line at a point.

Instantaneous rate of change

The rate of change of a function at a specific input value, given by the derivative at that point.

Leibniz notation (dy/dx)

Notation for the derivative of y with respect to x, emphasizing the dependent and independent variables.

Differential operator notation (d/dx)[f(x)]

Notation that emphasizes differentiation as an operation applied to a function f(x).

Prime notation (f'(x))

A common notation for the derivative of the function f with respect to x.

Linearity of derivatives

Property that derivatives distribute over addition/subtraction and pull out constant multiples.

Constant Rule

If f(x)=c (a constant), then f'(x)=0.

Constant Multiple Rule

If c is constant and f is differentiable, then d/dx[c·f(x)] = c·f'(x).

Sum Rule

If f and g are differentiable, then d/dx[f(x)+g(x)] = f'(x)+g'(x).

Difference Rule

If f and g are differentiable, then d/dx[f(x)-g(x)] = f'(x)-g'(x).

Power Rule

For real n (where defined), d/dx[x^n] = n·x^(n−1).

Polynomial

A function written as a sum of terms a·x^n where a is a constant and n is a nonnegative integer.

Rewriting with negative exponents

Strategy of converting fractions into powers (e.g., 5/x^3 = 5x^(−3)) to use the power rule instead of more complex rules.

Fractional exponents (radicals as exponents)

Writing roots as powers (e.g., √x = x^(1/2)) so the power rule can be applied.

Trig derivatives require radians

The standard trig derivative formulas (like (sin x)'=cos x) hold in their simple form when x is measured in radians.

Derivative of sin(x)

d/dx[sin(x)] = cos(x).

Derivative of cos(x)

d/dx[cos(x)] = −sin(x).

Derivatives of tan(x) and cot(x)

d/dx[tan(x)] = sec^2(x); d/dx[cot(x)] = −csc^2(x).

Derivatives of sec(x) and csc(x)

d/dx[sec(x)] = sec(x)tan(x); d/dx[csc(x)] = −csc(x)cot(x).

Derivative of e^x

d/dx[e^x] = e^x.

Derivative of a^x

For a>0, a≠1: d/dx[a^x] = a^x·ln(a).

Derivative of ln(x)

For x>0: d/dx[ln(x)] = 1/x.

Derivative of log_a(x)

For a>0, a≠1 and x>0: d/dx[log_a(x)] = 1/(x·ln(a)).

Product Rule

If h(x)=f(x)g(x), then h'(x)=f'(x)g(x)+f(x)g'(x).

Quotient Rule

If h(x)=f(x)/g(x) with g(x)≠0, then h'(x) = (f'(x)g(x) − f(x)g'(x)) / (g(x))^2.