The Handy Math Answer Book

This equation is also seen written as Dn(y) dny/dxn.

What is a partial derivative?

Partial derivatives (seen written as the symbol ) are derivatives of a function contain-
ing multiple variables that have all but the variable of interest held fixed during the
differentiation. Thus, when a function f(x, y,...) depends on more than one variable,
the partial derivative can be used to specify the derivative with respect to one or more
variables. There are other terms, too: Partial derivatives that involve more than one
variable are called mixed partial derivatives. And a differential equation expressing
one or more quantities in terms of partial derivatives is called, logically, a partial dif-
ferential equation. These equations are well known in physics and engineering, and
most are notoriously difficult to solve.

What are the product, quotient, power,and chain rulesfor derivatives?

There are numerous rules for derivatives of certain combinations of functions, includ-
ing the product, quotient, power, and chain rules. The following lists their common
notation:

For a product:

where f' is the derivative of f with respect to x.

For a quotient:

()

()

()

() () () ()

dx

d

gx

fx

gx

gxf x fxg x

= 2

ll-

<

6

F

@

dxd^6 fxgx() ()@=+fxg x f xgx() ()ll() ()

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MATHEMATICAL ANALYSIS

What is an example of the second derivative?

T

he second derivative is actually a function’s derivative’s derivative. In other

words, the function’s derivative may also have its own derivative, called the

second derivative or second order derivative. If we let yf(x), the second deriv-

ative becomes d/dx(dy/dx). This is equal to d^2 y/dx^2 , further represented by the

symbols f' (x) or y'. One good example of a second order derivative is accelera-

tion—it is actually the second derivative of a change in distance. In other words,

the first derivative gives instantaneous velocity (see above) while the second

derivative gives acceleration.