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 f(x) = e^h(x) or f(x) = sin(h(x)), the chain rule must be applied. Consider the outer function (exponentiation and sine in these examples) to be a function g(x). It follows that:

f'(x) = g'(h(x)) * h'(x)

Product Rule

Consider a function like f(x) = g(x)h(x). The derivative is evaluated as:

This product rule holds for vector multiplication; that is, for a dot product:

f = g ⋅ h

df/dx = h ⋅ (dg/dx) + g ⋅ (dh/dx)

...and also for a cross product:

f = g × h

df/dx = h × (dg/dx) + g × (dh/dx)

Quotient Rule

Consider a function like f(x) = g(x)/h(x). The derivative is evaluated 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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