Respuesta :
Answer:
-77.50
Step-by-step explanation:
[tex]5x^2+y^4=21[/tex]
We take first derivative
Derivative of x is 1. Derivative of y is dy/dx
Derivative of x^2 is 2x
Derivative of y^4 is 4y^3(dy/dx)
[tex]10x+4y^3(\frac{dy}{dx})=0[/tex]
Now isolate dy/dx
Subtract 10x on both sides
[tex]4y^3(\frac{dy}{dx})=-10x[/tex]
Divide both sides by 4y^3
[tex]\frac{dy}{dx}= \frac{-10x}{4y^3}[/tex]
[tex]\frac{dy}{dx}= \frac{-5x}{2y^3}[/tex]
Now we take second derivative. Apply quotient rule
Derivative of -5x is -5 and derivative of 2y^3 is 6y(dy/dx)
[tex]\frac{d^2y}{dx^2}= \frac{(-5)(2y^3)-(-5x)(6y^2\frac{dy}{dx})}{(2y^3)^2}[/tex]
Replace dy/dx , [tex]\frac{dy}{dx}= \frac{-5x}{2y^3}[/tex]
[tex]\frac{d^2y}{dx^2}= \frac{(-5)(2y^3)-(-5x)(6y^2\frac{-5x}{2y^3})}{(2y^3)^2}[/tex]
[tex]\frac{d^2y}{dx^2}= \frac{-10y^3-\frac{75x^2}{y}}{(4y^6)}[/tex]
Now plug in x=2 and y=1
[tex]\frac{d^2y}{dx^2}= \frac{-10(1)^3-\frac{75(2)^2}{1}}{(4(1)^6)}[/tex]
[tex]\frac{d^2y}{dx^2}= -77.50[/tex]
