Find the derivative of each of these functions. 16. $$ y=e^{x^{2}-1} $$ 17. $$ y=x^{4} \sin 3 x $$ 18. $$ f(x)=\cos (\sin 2 x) $$

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**16. Function: $$ y = e^{x^2-1} $$**

1. Identify the outer function as $$ e^u $$ with $$ u = x^2 - 1 $$.
2. Differentiate the outer function: $$\frac{d}{du}e^u = e^u$$.
3. Differentiate the inner function: $$\frac{d}{dx}(x^2-1) = 2x$$.
4. Apply the chain rule:
   $$
   y' = e^{x^2-1} \cdot 2x = 2x e^{x^2-1}
   $$

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**17. Function: $$ y = x^4 \sin 3x $$**

1. This is a product of two functions:
   - $$ u(x) = x^4 $$
   - $$ v(x) = \sin 3x $$
2. Differentiate $$ u(x) $$:
   $$
   u'(x) = 4x^3
   $$
3. Differentiate $$ v(x) $$ using the chain rule. With $$ w = 3x $$, we have:
   $$
   v(x) = \sin w \quad \text{and} \quad v'(x) = \cos w \cdot \frac{dw}{dx} = \cos 3x \cdot 3 = 3\cos 3x
   $$
4. Apply the product rule:
   $$
   y' = u'(x)v(x) + u(x)v'(x) = 4x^3 \sin 3x + x^4 (3\cos 3x)
   $$
5. Write the final answer:
   $$
   y' = 4x^3 \sin 3x + 3x^4 \cos 3x
   $$

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**18. Function: $$ f(x) = \cos (\sin 2x) $$**

1. Recognize the composition:
   - Outer function: $$ g(u) = \cos u $$
   - Inner function: $$ u(x) = \sin 2x $$
2. Differentiate the outer function:
   $$
   g'(u) = -\sin u
   $$
3. Differentiate the inner function using the chain rule. With $$ v = 2x $$, we have:
   $$
   u(x) = \sin v \quad \text{and} \quad u'(x) = \cos v \cdot \frac{dv}{dx} = \cos 2x \cdot 2 = 2\cos 2x
   $$
4. Apply the chain rule:
   $$
   f'(x) = g'(u(x)) \cdot u'(x) = -\sin (\sin 2x) \cdot 2\cos 2x
   $$
5. Write the final answer:
   $$
   f'(x) = -2\cos 2x\, \sin(\sin 2x)
   $$
**16.** The derivative of $$ y = e^{x^2-1} $$ is $$ y' = 2x e^{x^2-1} $$. **17.** The derivative of $$ y = x^4 \sin 3x $$ is $$ y' = 4x^3 \sin 3x + 3x^4 \cos 3x $$. **18.** The derivative of $$ f(x) = \cos (\sin 2x) $$ is $$ f'(x) = -2\cos 2x\, \sin(\sin 2x) $$.

Find the derivative of each of these functions. 16. \( y=e^{x^{2}-1} \) 17. \( y=x^{4} \sin 3 x \) 18. \( f(x)=\cos (\sin 2 x) \)

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