If yis held constant, then dy is zero and equation (12.35) yields the result
(
dz
dx
)
y
=−
∂F
∂x
∂F
∂z
(12 .36)
Here the symbol on the left of equation (12.36) is used to emphasize that yis being
held constant during the differentiation process. Some engineering texts feel this
notation is less ambiguous than the use of the partial derivative symbol occurring
in equation (12.34).
Differentiating the equation (12.32) with respect to yproduces the result
∂F
∂y
+∂F
∂z
∂z
∂y
= 0 (12 .37)
from which one finds the partial derivative
∂z
∂y
=−
∂F
∂y
∂F
∂z
, ∂F
∂z
= 0 (12 .38)
Alternatively, set x equal to a constant so that dx = 0 in equation (12.35), then
equation (12.35) produces the result
(
dz
dy
)
x
=−
∂F
∂y
∂F
∂z
,
∂F
∂z = 0 (12 .39)
Here the derivative is represented using the alternative notation
(
dz
dy
)
x
emphasizing
the derivative is obtained holding xconstant.
one equation, four unknowns
Any equation of the form
F(x, y, z, w ) = 0 (12 .40)
implicitly defines was one or more functions of x, y and z. Differentiate equation
(12.40) with respect to xand show
∂F
∂x
+∂F
∂w
∂w
∂x
= 0 or ∂w
∂x
=−
∂F
∂x
∂F
∂w
(12 .41)
Differentiate equation (12.40) with respect to yand show
∂F
∂y
+∂F
∂w
∂w
∂y
= 0 or ∂w
∂y
=−
∂F
∂y
∂F
∂w
(12 .42)
