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Partial derivatives

Partial derivatives

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6.8) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.

Partial derivatives 6.8. If f (x, y) = 2x? – xy + y², find (a) d fl d x and (b) d fld y at (xo, Yo) directly from the definition. af (a) ax f(x, + h, y,) – f(x,, Yo) = f,(x,, Yo) = lim h [2(x, + h)' – (x, + h)y, + y°]=[2x, – x,Yo + y°] = lim h 4hx, + 2h? – hy lim - 3 lim(4x, + 2h - y,) %3D 4x, — у h→0 h h0 af (b) ду f(Xp, Yo + k) – f(xp, Yo) = f,(x, Yo) = lim k [2x, – x,(Yo + k) + (y, + k)']– [2x, – x,Vo + y°] = lim k -kx, + 2ky, + k² = lim lim(-x, + 2y, + k)= – x, + 2y% k Since the limits exist for all points (Xp, Yo), we can write f,(x, y) =f;= 4x – y, f, (x, y) =fy=-x+2y, which are themselves functions of x and y. Note that formally f,(xo, Yo) is obtained from f(x, y) by differentiating with respect to x, keeping y constant and then putting x = Xp, y = Yo- Similarly, f,(Xo, Yo) is obtained by differentiating f with respect to y, keeping x constant. This procedure, while often lucrative in practice, need not always yield correct results (see Problem 6.9). It will work if the partial derivatives are continuous.
Transcribed Image Text:Partial derivatives 6.8. If f (x, y) = 2x? – xy + y², find (a) d fl d x and (b) d fld y at (xo, Yo) directly from the definition. af (a) ax f(x, + h, y,) – f(x,, Yo) = f,(x,, Yo) = lim h [2(x, + h)' – (x, + h)y, + y°]=[2x, – x,Yo + y°] = lim h 4hx, + 2h? – hy lim - 3 lim(4x, + 2h - y,) %3D 4x, — у h→0 h h0 af (b) ду f(Xp, Yo + k) – f(xp, Yo) = f,(x, Yo) = lim k [2x, – x,(Yo + k) + (y, + k)']– [2x, – x,Vo + y°] = lim k -kx, + 2ky, + k² = lim lim(-x, + 2y, + k)= – x, + 2y% k Since the limits exist for all points (Xp, Yo), we can write f,(x, y) =f;= 4x – y, f, (x, y) =fy=-x+2y, which are themselves functions of x and y. Note that formally f,(xo, Yo) is obtained from f(x, y) by differentiating with respect to x, keeping y constant and then putting x = Xp, y = Yo- Similarly, f,(Xo, Yo) is obtained by differentiating f with respect to y, keeping x constant. This procedure, while often lucrative in practice, need not always yield correct results (see Problem 6.9). It will work if the partial derivatives are continuous.
Expert Solution
Step 1

Given:

The function fx,y=2x2-xy+y2

To Find:

Using the definition,

a) ∂f∂x at x0, y0b) ∂f∂y at x0, y0

steps

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