If y = tan^(-1)x, find d²y/dx² in terms of y alone.

To solve the problem, we are given that y = tan^(-1)(x), and we are tasked with finding the second derivative of y with respect to x, expressed in terms of y alone.

Step 1: First, recall that the derivative of y = tan^(-1)(x) with respect to x is:

dy/dx = 1 / (1 + x^2).

Step 2: Now, we need to find the second derivative d²y/dx². To do this, we differentiate dy/dx with respect to x.

d²y/dx² = d/dx (1 / (1 + x^2)).

Using the quotient rule for differentiation:

d/dx (1 / (1 + x^2)) = (0 * (1 + x^2) - 1 * (2x)) / (1 + x^2)^2 = -2x / (1 + x^2)^2.

So, d²y/dx² = -2x / (1 + x^2)^2.

Step 3: To express the second derivative in terms of y alone, we know that y = tan^(-1)(x). From this, we can express x in terms of y:

x = tan(y).

Step 4: Now, substitute x = tan(y) into the expression for d²y/dx²:

d²y/dx² = -2(tan(y)) / (1 + (tan(y))^2)^2.

Step 5: Recall the trigonometric identity: 1 + tan^2(y) = sec^2(y). Using this identity:

d²y/dx² = -2(tan(y)) / (sec^2(y))^2 = -2(tan(y)) / sec^4(y).

Thus, the second derivative in terms of y is:

d²y/dx² = -2tan(y) / sec^4(y).