Basic differentiation rules

From definition to rules

The derivative definition is powerful, but using the limit every time would be slow. Differentiation rules summarize common patterns. These rules let you compute derivatives quickly while still relying on the meaning of derivatives as rates of change.

The most important habit is to connect each rule to interpretation. A derivative is not just a symbol operation. It tells how a quantity changes.

Constant rule

If a function is constant, it does not change. Therefore its derivative is zero:

rac{d}{dx}(c) = 0

For example, if $f(x)=7$, then $f'(x)=0$. The graph is horizontal, so its slope is zero everywhere.

In physics, a constant position has zero velocity, and a constant velocity has zero acceleration.

Power rule

The power rule is one of the most used rules:

rac{d}{dx}(x^n) = nx^{n-1}

For example:

rac{d}{dx}(x^2)=2x

rac{d}{dx}(x^3)=3x^2

rac{d}{dx}(sqrt{x}) = rac{d}{dx}(x^{1/2}) = rac{1}{2}x^{-1/2}

The power rule works for many powers, including negative and fractional powers where the function is defined.

Constant multiple and sum rules

If a function is multiplied by a constant, the derivative is multiplied by that same constant:

rac{d}{dx}[cf(x)] = crac{df}{dx}

If functions are added or subtracted, derivatives add or subtract:

rac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x)

These rules allow you to differentiate polynomials term by term.

For example, if

$f(x)=4x^3-2x+5$

then

$f'(x)=12x^2-2$

Exponential and logarithmic derivatives

The natural exponential has a special derivative:

rac{d}{dx}(e^x)=e^x

More generally,

rac{d}{dx}(e^{kx})=ke^{kx}

when combined with the chain rule.

The natural logarithm has derivative:

rac{d}{dx}ln x = rac{1}{x}

These derivatives appear in decay, growth, thermodynamics, waves, circuits, and probability.

Trigonometric derivatives

The basic trig derivatives are:

rac{d}{dx}sin x = cos x

rac{d}{dx}cos x = -sin x

These formulas assume $x$ is measured in radians. This is one of the reasons radians are essential in calculus-based physics.

For oscillation, if position is $x(t)=Acos(omega t)$, then velocity involves $-sin(omega t)$, and acceleration returns to a form proportional to $-cos(omega t)$.

Notation

Several derivative notations are common:

$f'(x)$

rac{df}{dx}

rac{d}{dx}f(x)

In physics, Leibniz notation such as $dx/dt$ is especially useful because it shows which variable is changing with respect to which.

Units of derivatives

A derivative's units are output units divided by input units. If position is measured in meters and time in seconds, $dx/dt$ has units $m/s$. If velocity is measured in $m/s$, then $dv/dt$ has units $m/s^2$.

Units help interpret derivatives physically.

The big idea

Differentiation rules are efficient tools for finding rates of change. Constants differentiate to zero, powers follow the power rule, sums differentiate term by term, and common exponential, logarithmic, and trig functions have standard derivatives. In physics, these rules turn mathematical functions into velocities, accelerations, growth rates, and response rates.