Calc I: Find the derivative: (6) $$ \frac{x+\sin (\cos (x))}{\cos \left(x^{2}\right)} $$ $$ y= $$

Question

UpStudy AI · Accepted Answer

We are given  
$$
y = \frac{x+\sin (\cos (x))}{\cos \left(x^{2}\right)}
$$  
and we want to find its derivative $$ y' $$.

**Step 1. Write the function as a quotient.**  
Let  
$$
f(x) = x+\sin(\cos(x)) \quad \text{and} \quad g(x)= \cos(x^2).
$$  
Then,  
$$
y = \frac{f(x)}{g(x)}.
$$

**Step 2. Compute $$ f'(x) $$.**  
Differentiate term by term:
- The derivative of $$ x $$ is $$ 1 $$.
- For $$ \sin(\cos(x)) $$, use the chain rule:
  $$
  \frac{d}{dx}\sin(\cos(x)) = \cos(\cos(x)) \cdot \frac{d}{dx}(\cos(x)) = \cos(\cos(x))\cdot (-\sin(x)) = -\sin(x)\cos(\cos(x)).
  $$
Thus,  
$$
f'(x) = 1 - \sin(x)\cos(\cos(x)).
$$

**Step 3. Compute $$ g'(x) $$.**  
Differentiate $$ g(x)=\cos(x^2) $$ using the chain rule:
- The derivative of $$ \cos(u) $$ with respect to $$ u $$ is $$ -\sin(u) $$.
- Here, $$ u=x^2 $$ so $$ u' = 2x $$.

Thus,  
$$
g'(x)= -\sin(x^2)\cdot 2x=-2x\sin(x^2).
$$

**Step 4. Apply the Quotient Rule.**  
The quotient rule states  
$$
y'=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}.
$$
Plug in $$ f(x) $$, $$ f'(x) $$, $$ g(x) $$, and $$ g'(x) $$:
$$
y'=\frac{\Bigl(1-\sin(x)\cos(\cos(x))\Bigr)\cos(x^2)-\Bigl[x+\sin(\cos(x))\Bigr]\bigl(-2x\sin(x^2)\bigr)}{\cos^2(x^2)}.
$$

**Step 5. Simplify the Expression.**  
Notice the negative sign in the second term:
$$
y'=\frac{\Bigl(1-\sin(x)\cos(\cos(x))\Bigr)\cos(x^2)+2x\sin(x^2)\Bigl[x+\sin(\cos(x))\Bigr]}{\cos^2(x^2)}.
$$

Thus, the derivative is:
$$
\boxed{y'=\frac{\left(1-\sin(x)\cos(\cos(x))\right)\cos(x^2)+2x\sin(x^2)\left[x+\sin(\cos(x))\right]}{\cos^2(x^2)}}.
$$
The derivative of $$ y = \frac{x + \sin(\cos(x))}{\cos(x^2)} $$ is: $$ y' = \frac{\left(1 - \sin(x)\cos(\cos(x))\right)\cos(x^2) + 2x\sin(x^2)\left[x + \sin(\cos(x))\right]}{\cos^2(x^2)} $$

Calc I: Find the derivative: (6) \( \frac{x+\sin (\cos (x))}{\cos \left(x^{2}\right)} \) \( y= \)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Mind Expander

preguntas relacionadas

Latest Calculus Questions