Question

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use the chain rule to evaluate the partial derivative dg/dtheta at the point (r,theta)=(2sqrt2, pi/4) where g(x,y)=1/6x+7y^2, x=rsin(theta), y=rcos(theta)

Answer 1

To find the partial derivative dg/dtheta, we use the given values of r and theta to substitute into x and y. Then, differentiate g(x, y) with respect to x and y. Finally, apply the chain rule to find dg/dtheta.

Step-by-step explanation:

To find the partial derivative dg/dtheta, we can use the chain rule. First, we substitute the given values of r and theta into x and y. x = rsin(theta) = 2sqrt(2)sin(pi/4) = 2, and y = rcos(theta) = 2sqrt(2)cos(pi/4) = 2sqrt(2).

Next, we differentiate g(x, y) = (1/6)x + 7y^2 with respect to x and y. dg/dx = 1/6 and dg/dy = 14y. Finally, we apply the chain rule: dg/dtheta = (dg/dx)(dx/dtheta) + (dg/dy)(dy/dtheta). Substituting the values we found earlier: dg/dtheta = (1/6)(0) + (14(2sqrt(2)))(-sqrt(2)) = -28.