The chain rule is a fundamental technique in calculus used for differentiating compositions of functions. If you have a function $y = f(g(x))$, where $f$ and $g$ are both differentiable functions, then the chain rule states:

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$
or

$$\frac{d}{dx} (f \circ g(x)) = f'(g(x)) \times g'(x)$$

or in Leibniz's notation:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

In simpler terms, you differentiate the outer function $f$ with respect to its inner argument $g(x)$, then multiply by the derivative of the inner function $g(x)$:

$$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$$

It also states that if you have a function $y$ that depends on $u$, which in turn depends on $x$, then the derivative of $y$ with respect to $x$ is:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Let the composite function be $y = (f \circ g)(x)$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{d}{dx} (f \circ g(x)) = f'(g(x)) \times g'(x)$$

Example of the Chain Rule

Let's look at a simple example: differentiate $h(x) = \sin(3x^2 + 2)$.

1. Identify the outer and inner functions:

   • Outer function $f(u) = \sin(u)$, where $u = g(x) = 3x^2 + 2$.

2. Differentiate the outer function $f$ with respect to $u$:

   • $f'(u) = \cos(u)$.

3. Differentiate the inner function $g(x) = 3x^2 + 2$:

   • $g'(x) = 6x$.

4. Apply the chain rule:

   • $h'(x) = f'(g(x)) \cdot g'(x) = \cos(3x^2 + 2) \cdot 6x$.

So, the derivative $h'(x) = 6x \cos(3x^2 + 2)$.