Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at that point. f(x,y)=y+sin(x/y), (0,3)

Answered question

2021-10-07

Explain why the function is differentiable at the given point. Then find the linearization L(x,y) of the function at  that point.

f(x,y)=y+sin(xy),(0,3)

Answer & Explanation

Jeffrey Jordon

Jeffrey Jordon

Expert2021-10-21Added 2605 answers

f(x,y)=y+sin(xy)

By the Sum Rule, the derivative of y+sin(xy)  with respect to y is ddy[y]+ddy[sin(xy)]

Differentiate using the Power Rule which states that ddy[yn] is ny1 where n=1

1+ddy[sin(xy)]

Evaluate ddy[sin(xy)]

Differentiate using the chain rule, which states that ddy[f(g(y))] is f(g(y))g(y) where f(y)=sin(y) and g(y)=xy

1+cos(xy)ddy[xy]

Since x is constant with respect to y, the derivative of xy with respect to y is xddy[1y]

1+cos(xy)(xddy[1y])

Rewrite 1y as y1

1+cos(xy)(xddy[y1])

 Differentiate using the Power Rule which states that ddy[yn] is ny1 where n=1

1+cos(xy)(x(y2))

Simplify.

xcos(xy)y2+1

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