Answer to Question #258535 in Calculus for Hamza

Question #258535

Derivative of x^3+4xy^2+5y^3=1 at (2,-1)


1
Expert's answer
2021-10-29T02:52:11-0400

Differentiate both sides with respect to xx


ddx(x3+4xy2+5y3)=ddx(1)\dfrac{d}{dx}(x^3+4xy^2+5y^3)=\dfrac{d}{dx}(1)

Use the Chain Rule


3x2+4y2+8xydydx+15y2dydx=03x^2+4y^2+8xy\dfrac{dy}{dx}+15y^2\dfrac{dy}{dx}=0

Solve for dydx\dfrac{dy}{dx}


dydx=3x2+4y28xy+15y2\dfrac{dy}{dx}=-\dfrac{3x^2+4y^2}{8xy+15y^2}

At (2,1)(2,-1)


dydx(2,1)=3(2)2+4(1)28(2)(1)+15(1)2=16\dfrac{dy}{dx}|_{(2,-1)}=-\dfrac{3(2)^2+4(-1)^2}{8(2)(-1)+15(-1)^2}=16

dydx(2,1)=16\dfrac{dy}{dx}|_{(2,-1)}=16


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