$$ \left. \begin{array} { l } { \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) } \ { \frac { d } { d x } ( \sin x ) ^ { 2 } } \ { = 2 \cos x } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { \frac { d } { d x } ( \sin ^ { 2 } x ^ { 2 } ) } \ { \frac { d } { d x } ( \sin x ) ^ { 2 } } \ { = 2 \cos x } \end{array} ight. $$

UpStudy AI · Accepted Answer

It appears you're attempting to differentiate two different functions and have provided the following expressions:

$$
\left. 
\begin{array}{l} 
\frac{d}{dx} \left( \sin^{2}x^{2} \right) \ 
\frac{d}{dx} \left( \sin x \right)^{2} \ 
= 2 \cos x 
\end{array} 
\right.
$$

However, there seems to be a misunderstanding in the differentiation process. Let's break down each derivative step by step to clarify:

---

### 1. Differentiating $$ \sin^{2}(x^{2}) $$

**Function:** $$ f(x) = \sin^{2}(x^{2}) $$

**Step-by-Step Differentiation:**

1. **Apply the chain rule:** The outer function is $$ u^2 $$ where $$ u = \sin(x^{2}) $$.

2. **Differentiate the outer function with respect to $$ u $$:**
   $$
   \frac{d}{du} u^{2} = 2u
   $$

3. **Differentiate the inner function $$ u = \sin(x^{2}) $$ with respect to $$ x $$:**
   $$
   \frac{d}{dx} \sin(x^{2}) = \cos(x^{2}) \cdot 2x = 2x \cos(x^{2})
   $$

4. **Combine the results using the chain rule:**
   $$
   \frac{d}{dx} \sin^{2}(x^{2}) = 2 \sin(x^{2}) \cdot 2x \cos(x^{2}) = 4x \sin(x^{2}) \cos(x^{2})
   $$

5. **Simplify using a trigonometric identity ($$ \sin(2\theta) = 2 \sin \theta \cos \theta $$):**
   $$
   4x \sin(x^{2}) \cos(x^{2}) = 2x \cdot 2 \sin(x^{2}) \cos(x^{2}) = 2x \sin(2x^{2})
   $$

**Final Result:**
$$
\frac{d}{dx} \sin^{2}(x^{2}) = 2x \sin(2x^{2})
$$

---

### 2. Differentiating $$ (\sin x)^{2} $$

**Function:** $$ g(x) = (\sin x)^{2} $$

**Step-by-Step Differentiation:**

1. **Apply the chain rule:** The outer function is $$ u^2 $$ where $$ u = \sin x $$.

2. **Differentiate the outer function with respect to $$ u $$:**
   $$
   \frac{d}{du} u^{2} = 2u
   $$

3. **Differentiate the inner function $$ u = \sin x $$ with respect to $$ x $$:**
   $$
   \frac{d}{dx} \sin x = \cos x
   $$

4. **Combine the results using the chain rule:**
   $$
   \frac{d}{dx} (\sin x)^{2} = 2 \sin x \cdot \cos x = 2 \sin x \cos x
   $$

5. **Simplify using a trigonometric identity ($$ \sin(2\theta) = 2 \sin \theta \cos \theta $$):**
   $$
   2 \sin x \cos x = \sin(2x)
   $$

**Final Result:**
$$
\frac{d}{dx} (\sin x)^{2} = \sin(2x)
$$

---

### 3. Addressing $$ = 2 \cos x $$

The derivative expressions you've provided are:

1. $$ \frac{d}{dx} \sin^{2}(x^{2}) = 2x \sin(2x^{2}) $$
2. $$ \frac{d}{dx} (\sin x)^{2} = \sin(2x) $$

Neither of these derivatives simplifies to $$ 2 \cos x $$. It seems there might have been a miscalculation or misunderstanding in applying the differentiation rules, particularly the chain rule and trigonometric identities.

**Key Points to Remember:**

- **Chain Rule:** When differentiating composite functions, multiply the derivative of the outer function by the derivative of the inner function.
  
- **Trigonometric Identities:** Utilize identities like $$ \sin(2\theta) = 2 \sin \theta \cos \theta $$ to simplify expressions.

If you have any specific questions or need further clarification on these differentiation steps, feel free to ask!
The derivatives are: 1. $$ \frac{d}{dx} \sin^{2}(x^{2}) = 2x \sin(2x^{2}) $$ 2. $$ \frac{d}{dx} (\sin x)^{2} = \sin(2x) $$ Neither equals $$ 2 \cos x $$.

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