If x^2=25-y^2, what is the value of [(d^2)(y)]\[dx^2] at the point (3,4)?
first derivative:
2x = -2y dy/dx
at (3,4)
dy/dx = -3/4
2nd derivative:
2 = -2y( d^2y/d^2x) + (-2)(dy/dx)
at the given point (3,4)
2 = -8(d^2 y/d^2 x) - 2(-3/4)
8 (d^2 y/d^2 x ) = 3/2 - 2 = -1/2
d^2 y/d^2 x = -1/16
To find the value of [(d^2)(y)]/[dx^2] at the point (3,4), we need to take the second derivative of y with respect to x and evaluate it at that point.
In the given equation x^2 = 25 - y^2, we can solve for y^2 and differentiate implicitly with respect to x:
x^2 + y^2 = 25
Differentiating both sides with respect to x gives:
2x + 2y(dy/dx) = 0
Rearranging and solving for dy/dx, we get:
dy/dx = -x/y
Now, we need to differentiate dy/dx with respect to x to find the second derivative [(d^2)(y)]/[dx^2]:
Using the quotient rule, we have:
[(d^2)(y)]/[dx^2] = [(d/dx)(-x/y)]/[dx]
To simplify, we differentiate both the numerator and denominator separately:
Numerator:
(d/dx)(-x/y) = (-1/y)(d/dx)(x) - (x)(d/dx)(1/y)
= (-1/y) - (x)(-1/y^2)(dy/dx)
= -1/y + (x/y^2)(dy/dx)
Denominator:
(dx/dx) = 1
Combining the numerator and denominator, we get:
[(d^2)(y)]/[dx^2] = (-1/y + (x/y^2)(dy/dx))/1
= -dy/dx/y + (x/y^2)(dy/dx)
Now, substitute the given point (3,4) into the expression to find the value of the second derivative:
At (3,4), x = 3 and y = 4. Also, we know from earlier that dy/dx = -x/y.
Substituting these values, we have:
[(d^2)(y)]/[dx^2] = -(-3)/4/4 + (3)/(4^2)(-3)/4
= 3/4 + 3/16
= 12/16 + 3/16
= 15/16
Therefore, the value of [(d^2)(y)]/[dx^2] at the point (3,4) is 15/16.