one equation, two unknowns
Any equation of the form
F(x, y ) = 0 (12 .28)implicitly defines y as one or more functions of x. Treating y as a function of x
differentiate the equation (12.28) with respect to xand show
∂F
∂x +∂F
∂ydy
dx = 0 (12 .29)from which one can solve for the first derivative to obtain
dy
dx =−∂F
∂x
∂F
∂y(12 .30)provided that ∂F∂y = 0. Higher derivatives are obtained by differentiating the first
derivative. For example, differentiate the equation (12.29) with respect to x and
show
∂^2 F
∂x^2+ ∂(^2) F
∂x ∂y
dy
dx
+∂F
∂y
d^2 y
dx^2
+dy
dx
[
∂^2 F
∂y ∂x
+∂
(^2) F
∂y^2
dy
dx
]
= 0 (12 .31)
One can then solve for the second derivative term. Higher ordered derivatives are
obtained by differentiating the equation (12.31).
one equation, three unknowns
An equation of the form
F(x, y, z) = 0 (12 .32)implicitly defines zas one or more functions of xand yprovided that ∂F∂z = 0.Treating
zas a function of xand yone can differentiate the equation (12.32) with respect to
xand obtain
∂F
∂x+∂F
∂z∂z
∂x= 0 (12 .33)Solving for ∂x∂z one finds
∂z
∂x=−∂F
∂x
∂F
∂z(12 .34)An alternative representation of the derivative of zwith respect to xcan be obtained
as follows. Take the differential of equation (12.32) to obtain
dF =∂F
∂x dx +∂F
∂y dy +∂F
∂z dz = 0 (12 .35)