Final answer:
The derivative of tan²(x²) with respect to x is 4x tan(x²) sec²(x²), which requires the use of the chain rule and known derivatives of trigonometric functions.
Step-by-step explanation:
The question asks for the derivative of the function tan2(x2).
To find this, we can use the chain rule and the fact that the derivative of tan(u) with respect to u is sec2(u).
Let's let u = x2 so that our function becomes (tan(u))2.
The derivative of u with respect to x is 2x.
Applying the chain rule, we find that the derivative of tan2(x2) with respect to x is 2x • 2 • tan(x2) • sec2(x2), which simplifies to 4x tan(x2) sec2(x2).
Therefore, the correct answer is the second option.