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 ($\frac{dy}{dx}$)

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

Differential operator notation ($\frac{d}{dx}[f(x)]$)

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 $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$.

Sum Rule

If $f$ and $g$ are differentiable, then $\frac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x)$.

Difference Rule

If $f$ and $g$ are differentiable, then $\frac{d}{dx}[f(x)-g(x)] = f'(x)-g'(x)$.

Power Rule

For real $n$ (where defined), $\frac{d}{dx}[x^n] = n \cdot x^{(n-1)}.$

Polynomial

A function written as a sum of terms $a \cdot x^n$ 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., $\sqrt{x} = x^{(1/2)}$) 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 $\sin(x)$

$\frac{d}{dx}[\sin(x)] = \cos(x)$.

Derivative of $\cos(x)$

$$\frac{d}{dx}[\cos(x)] = -\sin(x).$$

Derivatives of $\tan(x)$ and $\cot(x)$

$$\frac{d}{dx}[\tan(x)] = \sec^2(x); \frac{d}{dx}[\cot(x)] = -\csc^2(x).$$

Derivatives of $\sec(x)$ and $\csc(x)$

$$\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x); \frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x).$$

Derivative of $e^x$

$$\frac{d}{dx}[e^x] = e^x.$$

Derivative of $a^x$

For $a > 0$, $a \neq 1$: $\frac{d}{dx}[a^x] = a^x \cdot \ln(a).$

Derivative of $\ln(x)$

For $x > 0$: $\frac{d}{dx}[\ln(x)] = \frac{1}{x}.$

Derivative of $\log_a(x)$

For $a > 0$, $a \neq 1$ and $x > 0$: $\frac{d}{dx}[\log_a(x)] = \frac{1}{x \cdot \ln(a)}.$

Product Rule

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

Quotient Rule

If $h(x)=\frac{f(x)}{g(x)}$ with $g(x) \neq 0$, then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}.$