Derivative Rules (Power, Product, Quotient)

The Definition of the Derivative

The derivative of $f$ at $x$ is defined as:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

This limit gives the instantaneous rate of change of $f$ at $x$, and geometrically it represents the slope of the tangent line to the graph of $f$ at the point $(x, f(x))$.

While the definition is important conceptually, the derivative rules below let you compute derivatives efficiently without returning to the limit every time.

Basic Differentiation Rules

Constant Rule: If $f(x) = c$ (a constant), then $f'(x) = 0$.

Power Rule: If $f(x) = x^n$ for any real number $n$, then:

$f'(x) = nx^{n-1}$

This works for positive integers, negative integers, and fractions. Examples:

• $\frac{d}{dx}[x^5] = 5x^4$
• $\frac{d}{dx}[x^{-2}] = -2x^{-3}$
• $\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$

Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Together, these rules let you differentiate any polynomial term by term. For example:

$\frac{d}{dx}[3x^4 - 7x^2 + 5x - 9] = 12x^3 - 14x + 5$

The Product Rule

When two functions are multiplied, you cannot simply differentiate each factor separately. The product rule states:

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

A helpful mnemonic: "derivative of the first times the second, plus the first times the derivative of the second."

Example: Find $\frac{d}{dx}[x^3 \sin x]$.

$= 3x^2 \cdot \sin x + x^3 \cdot \cos x$

The Quotient Rule

For a quotient of two functions:

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}$

A common mnemonic: "lo d-hi minus hi d-lo, over lo-lo" (where "lo" = denominator, "hi" = numerator, and "d" = derivative of).

Example: Find $\frac{d}{dx}\left[\frac{x^2 + 1}{x - 3}\right]$.

$= \frac{2x(x-3) - (x^2+1)(1)}{(x-3)^2} = \frac{2x^2 - 6x - x^2 - 1}{(x-3)^2} = \frac{x^2 - 6x - 1}{(x-3)^2}$

Derivatives of Trigonometric Functions

You must memorize all six:

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\cot x$	$-\csc^2 x$
$\sec x$	$\sec x \tan x$
$\csc x$	$-\csc x \cot x$

Notice the pattern: the "co-" functions (cosine, cotangent, cosecant) all have a negative sign in their derivatives.

Derivatives of Exponential and Logarithmic Functions

While covered more deeply elsewhere, you should know:

$\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}$

$\frac{d}{dx}[a^x] = a^x \ln a \qquad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$

When to Use Product Rule vs. Quotient Rule

Sometimes you have a choice. For example, $\frac{x^3}{x^2}$ can be simplified to $x$ first — no quotient rule needed. Other times, rewriting a quotient as a product with a negative exponent lets you use the product rule instead:

$\frac{f(x)}{g(x)} = f(x) \cdot [g(x)]^{-1}$

The quotient rule and the product rule (with chain rule) will always give the same answer. Use whichever feels more natural for the given problem.

Tip 1: Simplify Before Differentiating

Always look for opportunities to simplify first. Expanding products, reducing fractions, or rewriting radicals as fractional powers can make differentiation much easier and reduce errors.

Tip 2: Rewrite Roots and Reciprocals as Powers

Convert $\sqrt{x}$ to $x^{1/2}$, $\frac{1}{x^3}$ to $x^{-3}$, and $\sqrt[3]{x^2}$ to $x^{2/3}$. This lets you apply the power rule directly.

Tip 3: Memorize Trig Derivatives Cold

You cannot afford to derive these from scratch during the exam. Flash-card drill until they are automatic. Remember the negative sign pattern for co-functions.

Tip 4: Check Your Work by Substituting a Value

After finding a derivative, plug in a specific $x$-value and verify against a numerical approximation using the limit definition: $f'(a) \approx \frac{f(a+0.001) - f(a)}{0.001}$.

Tip 5: Watch for Hidden Product/Quotient Structures

On the AP exam, functions are not always presented in obvious product or quotient form. Something like $x^2 \tan x$ requires the product rule, and $\frac{\sin x}{1 + \cos x}$ requires the quotient rule.