Final answer:
To find the second derivative of the equation y² = 4x, we first express y in terms of x and differentiate twice. The first derivative is dy/dx = 1/√x and the second derivative is d²y/dx² = -1/(2y³).
Step-by-step explanation:
Given the equation y² = 4x, to find the second derivative, we must first express y as a function of x. Taking the square root of both sides, we get y = 2√x, as the positive square root is typically taken for these types of problems. To find the first derivative, dy/dx, we apply the chain rule:
dy/dx = d/dx (2√x) = 2 · d/dx (√x) = 2 · (½)x^(-1/2) = 1/√x
Now to find the second derivative, d²y/dx², we differentiate dy/dx again:
d²y/dx² = d/dx (1/√x) = d/dx (x^(-1/2)) = -½ x^(-3/2)
To show this in terms of y, we can substitute back in for x using the original equation. Recall that x = y² / 4, so we have:
d²y/dx² = -½ (²y / 4)^(-3/2) = -½ (²y)^(-3/2) / (2³)
Therefore, the second derivative is -1/(2y³).