1. Identify inner ($u = x^2$) and outer ($y = \sin(u)$) functions. 2. Differentiate: $\frac{dy}{du} = \cos(u)$, $\frac{du}{dx} = 2x$. 3. Apply chain rule: $\frac{dy}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$.

1. Differentiate both sides: $2x + 2y\frac{dy}{dx} = 0$. 2. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = -\frac{x}{y}$.

1. Find the first derivative, $f'(x)$. 2. Find the derivative of $f'(x)$ to get the second derivative, $f''(x)$. 3. Repeat to find higher derivatives.

A function formed by applying one function to the results of another; $f(g(x))$.

A method to find $\frac{dy}{dx}$ when $y$ is not explicitly defined as a function of $x$.

A derivative of a derivative (e.g., second derivative, third derivative).

The derivative of a function, denoted as $f'(x)$ or $\frac{dy}{dx}$, representing the slope of the tangent line.

The derivative of the first derivative, denoted as $f''(x)$ or $\frac{d^2y}{dx^2}$, indicating the concavity of the function.

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

$\frac{d}{dx}(y^n) = n \cdot y^{n-1} \cdot \frac{dy}{dx}$

$\frac{f(b) - f(a)}{b - a}$