What is the Chain Rule formula (Leibniz notation)?
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
What is the Chain Rule formula (function notation)?
$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$
If $y = u^n$, what is $\frac{dy}{dx}$ using the Chain Rule?
$\frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx}$
If $y = e^{g(x)}$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = e^{g(x)} \cdot g'(x)$
If $y = \sin(g(x))$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = \cos(g(x)) \cdot g'(x)$
If $y = \cos(g(x))$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = -\sin(g(x)) \cdot g'(x)$
If $y = \ln(g(x))$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = \frac{1}{g(x)} \cdot g'(x) = \frac{g'(x)}{g(x)}$
If $y = [f(x)]^n$, what is $y'$?
$y' = n[f(x)]^{n-1} cdot f'(x)$
If $y = \tan(g(x))$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = \sec^2(g(x)) \cdot g'(x)$
If $y = \sqrt{g(x)}$, what is $\frac{dy}{dx}$?
$\frac{dy}{dx} = \frac{g'(x)}{2\sqrt{g(x)}}$
State the Chain Rule Theorem.
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 $F'(x) = f'(g(x)) \cdot g'(x)$.
What is the significance of the Chain Rule Theorem?
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.
How does the Chain Rule Theorem relate to the concept of differentiability?
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.
What conditions must be met to apply the Chain Rule Theorem?
$g$ must be differentiable at $x$, and $f$ must be differentiable at $g(x)$.
How is the Chain Rule Theorem used in related rates problems?
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.
How does the Chain Rule Theorem apply when differentiating inverse functions?
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.
Describe a scenario where the Chain Rule Theorem is essential.
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.
Explain how the Chain Rule Theorem ensures the correct derivative of a composite function.
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.
How does the Chain Rule Theorem relate to the concept of local linearity?
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.
Explain the connection between the Chain Rule Theorem and the derivative of a power function.
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 $y' = n[f(x)]^{n-1} \cdot f'(x)$.
Define a composite function.
A function formed by applying one function to the output of another: $(f \circ g)(x) = f(g(x))$.
What is the inner function in a composite function?
The function whose output serves as the input for the outer function, e.g., $g(x)$ in $f(g(x))$.
What is the outer function in a composite function?
The function that takes the output of the inner function as its input, e.g., $f(x)$ in $f(g(x))$.
What is the Chain Rule used for?
Differentiating composite functions.
State the Chain Rule in Leibniz notation.
$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
State the Chain Rule in function notation.
$\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$
What does $\frac{dy}{du}$ represent in the Chain Rule?
The derivative of the outer function with respect to the inner function.
What does $\frac{du}{dx}$ represent in the Chain Rule?
The derivative of the inner function with respect to $x$.
What is the first step in applying the Chain Rule?
Identify the inner and outer functions.
What is the second step in applying the Chain Rule?
Take the derivative of the outer function, leaving the inner function as is.
