Question

Find all the second order partial derivatives of g (x,y) = x Superscript 4 Baseline y + 3 sine (y) + 5 y cosine (x)

Answer 1

Answer:

Step-by-step explanation:

Given the function $g(x,y) = x^{4}y+3siny+5ycosx$

before we can get its second order partial derivative, we need to get its first order first. The first order are δg/δx and δg/δy

δg/δx  $= 4x^{3}y - 5ysinx$

δg/δy = $x^{4} +3cosy+5cosx$

The second derivatives are δ²g/δy²,  δ²g/δx², δ²g/ δyδx or δ²g/ δxδy

δ²g/δy² =  δ/δy (δg/δy) = δ/δy($x^{4}+3cosy+5cosx$)  

δ²g/δy² = -3siny

Similarly δ²g/δx² =  δ/δx (δg/δx) = δ/δx($4x^{3}y - 5ysinx$)

δ²g/δx² = 12x²y-5ycosx

δ²g/ δyδx =   δ/δy (δg/δx) =   δ/δy ($4x^{3}y - 5ysinx$)

δ²g/ δyδx  = 4x³ - 5sinx = δ²g/ δxδy ( for continuous function)