The chain rule is a fundamental technique in calculus used to differentiate compositions of functions. It states that if you have a composite function \( f(g(x)) \), the derivative of this function with respect to \( x \) is given by:

\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \cdot g'(x) \]

In other words, you first differentiate the outer function \( f \) with respect to its argument \( g(x) \), and then multiply by the derivative of the inner function \( g \) with respect to \( x \).

### Example 1: Differentiating \( \sin(x^2) \)

Let's find the derivative of \( h(x) = \sin(x^2) \).

1. **Identify the outer and inner functions:**
   - Outer function \( f(u) = \sin(u) \), where \( u = x^2 \).
   - Inner function \( g(x) = x^2 \).

2. **Differentiate the outer function:**
   \[ f'(u) = \cos(u) \]

3. **Differentiate the inner function:**
   \[ g'(x) = 2x \]

4. **Apply the chain rule:**
   \[ \frac{d}{dx} [\sin(x^2)] = \cos(x^2) \cdot 2x \]

So, the derivative of \( \sin(x^2) \) is \( 2x \cos(x^2) \).

### Example 2: Differentiating \( e^{\tan(x)} \)

Let's find the derivative of \( k(x) = e^{\tan(x)} \).

1. **Identify the outer and inner functions:**
   - Outer function \( f(u) = e^u \), where \( u = \tan(x) \).
   - Inner function \( g(x) = \tan(x) \).

2. **Differentiate the outer function:**
   \[ f'(u) = e^u \]

3. **Differentiate the inner function:**
   \[ g'(x) = \sec^2(x) \]

4. **Apply the chain rule:**
   \[ \frac{d}{dx} [e^{\tan(x)}] = e^{\tan(x)} \cdot \sec^2(x) \]

So, the derivative of \( e^{\tan(x)} \) is \( e^{\tan(x)} \sec^2(x) \).

### Example 3: Differentiating \( \ln(3x^2 + 1) \)

Let's find the derivative of \( m(x) = \ln(3x^2 + 1) \).

1. **Identify the outer and inner functions:**
   - Outer function \( f(u) = \ln(u) \), where \( u = 3x^2 + 1 \).
   - Inner function \( g(x) = 3x^2 + 1 \).

2. **Differentiate the outer function:**
   \[ f'(u) = \frac{1}{u} \]

3. **Differentiate the inner function:**
   \[ g'(x) = 6x \]

4. **Apply the chain rule:**
   \[ \frac{d}{dx} [\ln(3x^2 + 1)] = \frac{1}{3x^2 + 1} \cdot 6x \]

So, the derivative of \( \ln(3x^2 + 1) \) is \( \frac{6x}{3x^2 + 1} \).

Would you like more details or have any questions about this topic? Here are some related questions you might find interesting:

1. What is the chain rule for higher-order derivatives?
2. How do you differentiate \( \cos(\ln(x)) \) using the chain rule?
3. What is the derivative of \( \sqrt{x^2 + 1} \) using the chain rule?
4. How do you apply the chain rule to implicit differentiation?
5. Can you explain the chain rule in the context of multivariable calculus?

**Tip:** When applying the chain rule, carefully identify the outer and inner functions, and take each derivative step-by-step to avoid mistakes.

Understand the chain rule in calculus with clear examples and step-by-step solutions. Learn how to differentiate composite functions such as sin(x^2), e^(tan(x)), and ln(3x^2 + 1). Suitable for undergraduate students and those preparing for calculus exams.

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