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Home Partial Differentiation Total Differentials and Tangent Planes Examples Example 2: Increment and Total Differential (Biquadratic Function)
 
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Example 2: Increment and Total Differential (Biquadratic Function)

Find the increment and total differential of z = x2 - 3xy2.

Increment:

Δz = [(x + Δx)2 - 3(x + Δx)(y + Δy)2] - [x2 - 3xy2]
= [x2 + 2x Δx + Δx2 - 3xy2 - 6xy Δy - 3x Δy2 - 3 Δxy2 - 6 Δxy Δy - 3 Δx Δy2] - [x2 - 3xy2]
= 2x - Δx + Δx2 - 6xy Δy - 3x Δy2 - 3 Δxy2 - 6 Δxy Δy - 3 Δx Δy2
= (2x - 3y2) Δx - 6xy Δy + Δx2 - 3x Δy2 - 6y Δx Δy - 3 Δx Δy2.

Total differential :

11_partial_differentiation-221.gif = 2x - 3y2, 11_partial_differentiation-222.gif= -6xy.

dz = 11_partial_differentiation-223.gif = (2x - 3y2) dx - 6xy dy.
Home Partial Differentiation Total Differentials and Tangent Planes Examples Example 2: Increment and Total Differential (Biquadratic Function)

Last Update: 2010-11-25