Final answer:
To find the second derivative of the equation 3x² - 4y² = 2x with respect to x, one must implicitly differentiate the equation twice and apply algebraic manipulation to isolate y''.
Step-by-step explanation:
To find the second derivative of 3x² - 4y² = 2x with respect to x, we must first take the derivative implicitly of both sides of the equation with respect to x.
Starting with the original equation:
3x² - 4y² = 2x
Differentiating both sides with respect to x, we get:
6x - 8yy' = 2
Solving for y', we find:
y' = ³(d2y)/(dx2)
Now, we differentiate the expression for y' again with respect to x in order to find the second derivative:
6 - 8(y'y + yy'') = 0
Since y' is a function of x, we need to apply the product rule when differentiating it:
-8(y'y + yy'') = -6
Rearranging for yy'', we obtain:
yy'' + y'2 = -¾
Note that this process involves the chain rule, product rule, and the implicit differentiation technique. You can solve for y'' (the second derivative of y with respect to x) from here, assuming you have the expression for y' from the first derivative step.