Basic Properties and Formulas

Basic Derivative Rules

Constant Rule: The derivative of a constant is zero. $\frac{d}{dx} c = 0$
Constant Multiple Rule: The derivative of a constant multiplied by a function is the constant times the derivative of the function. $\frac{d}{dx} [c f(x)] = c f'(x)$
Power Rule: The derivative of a variable raised to a power is the power multiplied by the variable raised to one less than the power. $\frac{d}{dx} x^n = nx^{n-1}$

The derivative of a sum of two functions is the sum of their derivatives.

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

The derivative of a difference between two functions is the difference of their derivatives.

\frac{d}{dx} [f(x) - g(x)] = f'(x) - g'(x)

The derivative of the product of two functions is given by:

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

The derivative of the quotient of two functions is:

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

For two functions $g$ and $h$ , with $y = g(u)$ and $u = h(x)$ , the derivative of $y$ with respect to $x$ is:

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

Using Lagrange's notation, where $'$ denotes the derivative, for $y = g(h(x))$ , the chain rule is expressed as:

y'(x) = g'(h(x)) \cdot h'(x)

For compositions involving multiple functions, such as $f = f \circ g \circ h$ , the chain rule extends as follows:

\frac{df}{dt} = \frac{df}{dg} \cdot \frac{dg}{dh} \cdot \frac{dh}{dt}

Or, using Lagrange notation for a function $f(t)$ that is a composition of $f$ , $g$ , and $h$ , we get:

(f \circ g \circ h)'(t) = f'(g(h(t))) \cdot g'(h(t)) \cdot h'(t)

Derivative of a Square: The derivative of the square of a function is twice the function times the derivative of the function. $\frac{d}{dx} [f(x)]^2 = 2 f(x) f'(x)$
Derivative of a Reciprocal: The derivative of the reciprocal of a function. $\frac{d}{dx} \left( \frac{1}{g(x)} \right) = -\frac{g'(x)}{[g(x)]^2}$