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

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

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

Differentiate the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

Differentiate both sides of the equation with respect to $x$, applying the chain rule to terms involving $y$, and solve for $\frac{dy}{dx}$.

The concavity of the original function and helps find points of inflection.

The slope of the original function.

The derivative at a point is the slope of the tangent line at that point.

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.