The chain rule is a fundamental rule in calculus for differentiating composite functions. It allows you to differentiate a function that is made up of another function, i.e., when one function is nested inside another.

Formal Statement of the Chain Rule:

If you have two functions, $f(x)$ and $g(x)$, and the composition of these functions is $y = f(g(x))$, then the derivative of $y$ with respect to $x$ is:

$\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}$

This means you differentiate the outer function $f$ with respect to $g(x)$, then multiply by the derivative of the inner function $g(x)$ with respect to $x$.

Step-by-Step Process:

1. Differentiate the outer function $f$ with respect to $g(x)$.
2. Multiply by the derivative of the inner function $g(x)$.

Example 1:

Given $y = \sin(3x^2)$, here the outer function is $\sin(u)$ and the inner function is $u = 3x^2$.

1. Differentiate the outer function $\sin(u)$ with respect to $u$: $\frac{d}{du}(\sin(u)) = \cos(u)$

2. Differentiate the inner function $3x^2$ with respect to $x$: $\frac{d}{dx}(3x^2) = 6x$

3. Apply the chain rule: $\frac{dy}{dx} = \cos(3x^2) \cdot 6x = 6x \cos(3x^2)$

Example 2:

For $y = e^{5x^3}$, let’s break it down:

• Outer function: $e^u$ where $u = 5x^3$
• Inner function: $u = 5x^3$

1. Differentiate the outer function $e^u$ with respect to $u$: $\frac{d}{du}(e^u) = e^u$

2. Differentiate the inner function $5x^3$ with respect to $x$: $\frac{d}{dx}(5x^3) = 15x^2$

3. Apply the chain rule: $\frac{dy}{dx} = e^{5x^3} \cdot 15x^2 = 15x^2 e^{5x^3}$

Tip:

When applying the chain rule, always carefully identify the outer and inner functions. A common mistake is to differentiate them in the wrong order or forget to chain them together.

Would you like to explore more examples or need further clarification? Here's a list of related topics you can explore:

1. The power rule in differentiation
2. The product rule in differentiation
3. The quotient rule in differentiation
4. Implicit differentiation
5. Higher-order derivatives and their applications