Respuesta :
Answer:
A. -25/27
Step-by-step explanation:
Given:
The equation is given as:
[tex]x^2+y^2=25[/tex]
To find: [tex]\frac{d^2 y}{dx^2}[/tex] at (4, 3)
Differentiating the above equation with respect to 'x', we get:
[tex]\frac{d}{dx}(x^2+y^2)=\frac{d}{dx}(25)\\\\2x+2yy'=0\\\\x+yy'=0\\\\yy'=-x\\\\y'=\frac{-x}{y}------- (1)[/tex]
Value of [tex]y'[/tex] at (4,3) is given as:
[tex]y'_{(4,3)}=-\dfrac{4}{3}-------- (2)[/tex]
Now, differentiating equation (1) with respect to 'x' again, we get:
[tex]y''=\frac{d}{dx}(\frac{-x}{y})\\\\y''=\frac{y(-1)-(-x)y'}{y^2}\\\\y''=\frac{-y+xy'}{y^2}[/tex]
Now, value of [tex]y''[/tex] at (4,3) is given as by plugging 4 for 'x', 3 for 'y' and [tex]\frac{-4}{3}[/tex] for [tex]y'[/tex]
[tex]y''_{(4,3)}=\frac{-3+(4)(-\frac{4}{3})}{3^2}\\\\y''_{(4,3)}=\frac{-3-\frac{16}{3}}{9}\\\\y''_{(4,3)}=\frac{-9-16}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\div 9\\\\y''_{(4,3)}=\frac{-25}{3}\times \frac{1}{9}\\\\y''_{(4,3)}=-\frac{25}{27}[/tex]
Therefore, the value of the second derivative at (4, 3) is option (A) which is equal to -25/27.
