If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then the composite function $F(x) = f(g(x))$ is differentiable at $x$ and $F'(x)$ is given by $F'(x) = f'(g(x)) \cdot g'(x)$.

It provides a formal justification for the method of differentiating composite functions, ensuring that the derivative is calculated correctly by accounting for the rates of change of both the inner and outer functions.

The theorem requires that both the inner function $g(x)$ and the outer function $f(x)$ be differentiable at their respective points for the composite function to be differentiable.

$g$ must be differentiable at $x$, and $f$ must be differentiable at $g(x)$.

It allows us to relate the rates of change of different variables in an equation by differentiating with respect to time and applying the Chain Rule to each term, ensuring that all rates of change are properly accounted for.

If $f$ and $g$ are inverse functions, then $f(g(x)) = x$. Differentiating both sides using the Chain Rule gives $f'(g(x)) \cdot g'(x) = 1$, which can be used to find the derivative of the inverse function.

When dealing with complex functions involving multiple layers of composition, such as $y = \sin(e^{x^2})$, the Chain Rule Theorem is essential for systematically differentiating each layer and obtaining the correct derivative.

It ensures the correct derivative by accounting for the rates of change of both the inner and outer functions and multiplying them together, providing a complete picture of how the composite function changes with respect to its input.

The Chain Rule Theorem relies on the concept of local linearity, where differentiable functions can be approximated by linear functions at a point, allowing us to break down the differentiation of composite functions into smaller, linear approximations.

When differentiating a power function where the base is a function of $x$, such as $y = [f(x)]^n$, the Chain Rule Theorem is used to account for the derivative of the base, resulting in $y' = n[f(x)]^{n-1} \cdot f'(x)$.

It allows us to differentiate composite functions, which are functions within functions, by breaking down the differentiation process into smaller, manageable steps.

Differentiate the outermost function first, treating the inner function as a single variable, then multiply by the derivative of the inner function.

It's often used in conjunction with other rules like the Product Rule and Quotient Rule when dealing with more complex functions.

You will get an incorrect derivative, as you've only differentiated the outer function and not accounted for the inner function's rate of change.

The inner function is the one 'inside' another function, while the outer function is the one that 'encloses' the inner function.

The Chain Rule is specifically designed to differentiate composite functions, so understanding their structure is essential for applying the rule correctly.

Apply the Chain Rule iteratively, starting with the outermost function and working your way inwards, multiplying by the derivative of each inner function at each step.

Forgetting to apply the chain rule to every layer of the function. Don't forget the derivative of the innermost function!

The Chain Rule is a fundamental tool in implicit differentiation, where we differentiate equations that are not explicitly solved for one variable in terms of the other.

It allows us to relate the rates of change of different variables in an equation by differentiating with respect to time and applying the Chain Rule to each term.

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

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

$\frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx}$

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

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

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

$\frac{dy}{dx} = \frac{1}{g(x)} \cdot g'(x) = \frac{g'(x)}{g(x)}$

$y' = n[f(x)]^{n-1} cdot f'(x)$

$\frac{dy}{dx} = \sec^2(g(x)) \cdot g'(x)$

$\frac{dy}{dx} = \frac{g'(x)}{2\sqrt{g(x)}}$