Explicit functions have 'y' isolated. Implicit functions have 'x' and 'y' mixed, requiring implicit differentiation.

It reveals explicit functions representing parts of the implicit relation and helps determine domain and range.

Positive slope: 'y' increases as 'x' increases. Negative slope: 'y' decreases as 'x' increases. Zero slope: horizontal tangent. Undefined slope: vertical tangent.

The function has a local maximum or minimum, or a point of inflection where the slope is momentarily zero.

The derivative is undefined, often indicating a cusp or a point where the function changes direction sharply.

Differentiate both sides with respect to x, apply the chain rule where necessary, and solve for dy/dx.

The domain of the resulting explicit function(s) may be narrower than the original implicit equation's allowed x-values.

Find dy/dx using implicit differentiation, evaluate at the given point to find the slope, and use the point-slope form of a line.

Related rates problems involve finding the rate at which one quantity is changing by relating it to other quantities whose rates of change are known, often using implicit differentiation.

When differentiating terms involving 'y' with respect to 'x', the chain rule requires multiplying by $\frac{dy}{dx}$ because 'y' is a function of 'x'.

Explicit: Differentiate directly. Implicit: Differentiate with respect to x, using chain rule for y terms, then solve for dy/dx. Explicit is easier when possible.

Horizontal: dy/dx = 0. Vertical: dy/dx is undefined. Horizontal indicates a local extremum or inflection point. Vertical indicates a cusp or sharp turn.

Solving for x: expresses x as a function of y. Solving for y: expresses y as a function of x. The choice depends on which is easier and what information you need.

Explicit: Chain rule applied when differentiating a composite function directly. Implicit: Chain rule always applied when differentiating y terms with respect to x.

The domain of the explicit solution can be more restricted than the implicit equation due to square roots or other restrictions.

Explicit: Rates are usually directly given. Implicit: Rates are related through an equation, requiring implicit differentiation with respect to time.

Explicit: Quotient rule used when differentiating a quotient of functions directly. Implicit: Quotient rule may be needed when solving for $\frac{dy}{dx}$ after implicit differentiation.

Explicit: Often simpler to visualize directly. Implicit: Can represent more complex relationships, but may require more analysis to graph.

Implicit: Differentiate $x^2+y^2=r^2$ directly. Explicit: Solve for y first, then differentiate. Implicit is often easier.

Explicit: Find dy/dx directly, then use point-slope form. Implicit: Find dy/dx implicitly, then use point-slope form. Implicit is necessary if you cannot solve for y.

$x^2 + y^2 = r^2$

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

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

$\frac{dy}{dx} = -\frac{x}{y}$

$y = \pm \frac{1}{2}\sqrt{16 - x^2}$

$m = \frac{dy}{dx}$

$\frac{d}{dx}f(x,y) = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} \frac{dy}{dx}$

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

$y - y_0 = \frac{dy}{dx}|_{(x_0, y_0)} (x - x_0)$

$y = \pm \sqrt{1-x^2}$