Question

1. - / 5.26 points find the derivative of the function. y = cot²(cos θ) y = resources read it watch it submit answer 11. - / 5.26 points find the derivative of the function. y = sin(sin(sin x)) y = resources read it

Explanation:

Step1: Differentiate outer function

Let $u = \cot(\cos\theta)$, so $y = u^2$. Then $y' = 2u \cdot u' = 2\cot(\cos\theta) \cdot \frac{d}{d\theta}[\cot(\cos\theta)]$

Step2: Differentiate cotangent term

$\frac{d}{d\theta}[\cot(\cos\theta)] = -\csc^2(\cos\theta) \cdot \frac{d}{d\theta}[\cos\theta]$

Step3: Differentiate cosine term

$\frac{d}{d\theta}[\cos\theta] = -\sin\theta$

Step4: Combine results

$y' = 2\cot(\cos\theta) \cdot [-\csc^2(\cos\theta) \cdot (-\sin\theta)] = 2\sin\theta \cot(\cos\theta) \csc^2(\cos\theta)$

Step5: Differentiate outer sine

Let $v = \sin(\sin x)$, so $y = \sin v$. Then $y' = \cos v \cdot v' = \cos(\sin(\sin x)) \cdot \frac{d}{d x}[\sin(\sin x)]$

Step6: Differentiate middle sine

Let $w = \sin x$, so $v = \sin w$. Then $\frac{d}{d x}[\sin(\sin x)] = \cos w \cdot w' = \cos(\sin x) \cdot \cos x$

Step7: Combine results for Q11

$y' = \cos(\sin(\sin x)) \cdot \cos(\sin x) \cdot \cos x$

Answer:

$2\sin\theta \cot(\cos\theta) \csc^2(\cos\theta)$
$\cos(\sin(\sin x))\cos(\sin x)\cos x$