Derivative

A derivative is an instantaneous rate of change with respect to an input variable. It is a ratio of differentials.

Rules

The basic rules/identities are:

For trigonometric functions:

Chain Rule

For composite functions like e^2x or sin(2x), the chain rule must be applied. Let f be the entire function as-is (e.g., e^2x), h be the 'inner function (e.g., 2x), and g be the 'outer' function (e.g., e^h(x)).

An inconvenient function h(x) can be rewritten as e^ln(h(x)), which can be evaluated using this rule.

Product Rule

Consider a function like f(x) = g(x)h(x). Evaluate as:

This product rule holds for vector multiplication. That is, given a vector f that was defined as a dot product like f = g ⋅ h, the derivative (i.e., over time) of f can be calculated from known derivatives of the components. In this case, try:

The same is true of an f defined as a cross product like f = g × h. Try:

Quotient Rule

Consider a function like f(x) = g(x)/h(x). Evaluate as:

Properties

Derivatives are linear: given a function defined like f(x) = αg(x) + βh(x):

.

The follows from the total differential; substitute g and h for g(x) and h(x):

f = gh

df = f_gdg + f_hdh

df/dx = f_g(dg/dx) + f_h(dh/dx)

And clearly the partial derivatives f_g and f_h are equal to h and g respectively, giving:

df/dx = h(dg/dx) + g(dh/dx)

Substituting back in the original functions gives the product rule.

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