To find the second derivative of y with respect to x, we first find the first derivative of y:
dy/dx = sec^2 x + tan x
Next, we take the derivative of this expression with respect to x to find the second derivative:
d2y/dx2 = d/dx (sec^2 x + tan x) = 2sec x sec x tan x + sec^2 x
Now we use the identity sec^2 x – tan^2 x = 1 to write sec^2 x = tan^2 x + 1, and substitute this into our expression for the second derivative:
d2y/dx2 = 2sec x (tan^2 x + 1) tan x + (tan^2 x + 1) = 2tan x sec x + 2sec x tan^3 x + 1
Next, we use the identity tan x = sin x / cos x and simplify the expression:
d2y/dx2 = 2(sin x / cos x)(1 / cos x) + 2(1 / cos x)(sin^3 x / cos x) + 1 = 2sin x / cos^2 x + 2sin^3 x / cos^3 x + 1 / cos^2 x
We can then simplify this expression further by using the identity sin^2 x + cos^2 x = 1 to write sin^2 x = 1 – cos^2 x:
d2y/dx2 = 2sin x / cos^2 x + 2sin^3 x / cos^3 x + 1 / cos^2 x = 2sin x / cos^2 x + 2(sin^2 x)(sin x / cos^3 x) + 1 / cos^2 x = 2sin x / cos^2 x + 2(cos^2 x – 1)(sin x / cos^3 x) + 1 / cos^2 x = 2sin x / cos^2 x + 2sin x / cos x – 2 / cos^2 x + 1 / cos^2 x = 2sin x / cos^2 x + 2sin x / cos x – 1 / cos^2 x
Finally, we use the identity 1 – sin^2 x = cos^2 x to write 1 / cos^2 x = 1 + tan^2 x, and substitute this into our expression:
d2y/dx2 = 2sin x / cos^2 x + 2sin x / cos x – 1 / cos^2 x = 2sin x / cos^2 x + 2sin x / cos x – 1 – tan^2 x = 2sin x / cos^2 x + 2sin x / cos x – sec^2 x
We can then simplify this expression further by using the identity sin x = (1 / cos x)sin x, and factoring out (1 / cos x):
d2y/dx2 = 2sin x / cos^2 x + 2sin x / cos x – sec^2 x = 2sin x (1 + cos x) / cos^3 x – 1 / cos^2 x = (2sin x + 2sin x cos x – 1) / cos^3 x
Next, we use the identity 1 – sin x cos x = sin^2 x / (1 + cos x) to write 2sin x cos x = 2sin
