The Chain Rule For Partial Derivatives - HP 50g User Manual

Graphing calculator.

Third-, fourth-, and higher order derivatives are defined in a similar manner.
To calculate higher order derivatives in the calculator, simply repeat the
derivative function as many times as needed. Some examples are shown
below:

The chain rule for partial derivatives

Consider the function z = f(x,y), such that x = x(t), y = y(t). The function z
actually represents a composite function of t if we write it as z = f[x(t),y(t)]. The
chain rule for the derivative dz/dt for this case is written as
To see the expression that the calculator produces for this version of the chain
rule use:
The result is given by d1y(t) d2z(x(t),y(t))+d1x(t) d1z(x(y),y(t)). The term d1y(t)
is to be interpreted as "the derivative of y(t) with respect to the 1
variable, i.e., t", or d1y(t) = dy/dt. Similarly, d1x(t) = dx/dt. On the other
hand, d1z(x(t),y(t)) means "the first derivative of z(x,y) with respect to the first
independent variable, i.e., x", or d1z(x(t),y(t)) = z/ x. Similarly, d2z(x(t),y(t))
= z/ y. Thus, the expression above is to be interpreted as:
z
v
dz/dt = (dy/dt) ( z/ y) + (dx/dt)
z
x
z
y
x
v
y
v
st
independent
z/ x).
(
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