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.