Find the derivative of the given function.
$y=\tan \left(e^{-x^3}\right)$

• A. $\(\sec\)^2\(\left\)(e^{-x^3}\(\right\))$
• B. $e^{-x^3}\(\cdot\]\sec\)^2\(\left\)(e^{-x^3}\(\right\))$
• C. $-3x^2\(\cdot\) e^{-x^3}\(\cdot\]\sec\)^2\(\left\)(e^{-x^3}\(\right\))$
• D. $-x^3\(\cdot\) e^{-x^3-1}\(\cdot\]\sec\)^2\(\left\)(e^{-x^3}\(\right\))$

Answer: (C)

Verified step by step guidance

1. Identify the function y = \(\tan\)\(\left\)(e^{-x^3}\(\right\)). We need to find its derivative with respect to x.
2. Recognize that this is a composite function, where the outer function is \(\tan\)(u) and the inner function is u = e^{-x^3}. We will use the chain rule to differentiate.
3. Differentiate the outer function \(\tan\)(u) with respect to u, which gives \(\sec\)^2(u).
4. Differentiate the inner function u = e^{-x^3} with respect to x. This requires using the chain rule again: differentiate e^{-x^3} to get -3x^2 \(\cdot\) e^{-x^3}.
5. Combine the derivatives using the chain rule: multiply \(\sec\)^2(e^{-x^3}) by the derivative of the inner function, -3x^2 \(\cdot\) e^{-x^3}, to get the derivative of the original function.