If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.

If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.

A function can only change from increasing to decreasing (or vice versa) at critical points.

Choose a test point in each interval defined by critical points, evaluate $f'(x)$ at that point. The sign of $f'(x)$ indicates whether $f(x)$ is increasing or decreasing on that interval.

It indicates a critical point where the function may have a local maximum, local minimum, or a point of inflection.

Evaluate the derivative of the function at that point. If the derivative is positive, the function is increasing; if negative, it's decreasing.

A positive derivative means the tangent line has a positive slope (increasing function), and a negative derivative means the tangent line has a negative slope (decreasing function).

Because the function's behavior can change at these points, even though they are not critical points in the traditional sense.

The first derivative test uses the sign changes of the first derivative around a critical point to determine if it's a local maximum or minimum.

Create a sign chart with critical points, test values, and the sign of the derivative. Positive signs indicate increasing intervals, and negative signs indicate decreasing intervals.

$\frac{d}{dx}(x^n) = nx^{n-1}$

$\frac{d}{dx}(c) = 0$

$\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x)$

$\frac{d}{dx}[cf(x)] = cf'(x)$

$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

\frac{d}{dx}[rac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

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

$\frac{d}{dx}(\sin(x)) = \cos(x)$

$\frac{d}{dx}(\cos(x)) = -\sin(x)$

$\frac{d}{dx}(e^x) = e^x$

1. Find $f'(x)$. 2. Find critical points (where $f'(x) = 0$ or is undefined). 3. Create intervals using critical points. 4. Choose test points in each interval. 5. Evaluate $f'(x)$ at each test point. 6. Determine if $f(x)$ is increasing ($f'(x) > 0$) or decreasing ($f'(x) < 0$).

1. Find the derivative $f'(x)$. 2. Set $f'(x) = 0$ and solve for $x$. 3. Identify $x$ values where $f'(x)$ is undefined.

Select any value within each interval created by the critical points. The test point should be easy to evaluate in the derivative.

Include the point of discontinuity when creating intervals, as the function's behavior can change there.

The function is always increasing on its domain.

The function is always decreasing on its domain.

Find the derivative of each piece of the function, determine critical points within each piece, and analyze the sign of the derivative in each interval, considering the points where the pieces connect.

Use implicit differentiation to find $\frac{dy}{dx}$, set $\frac{dy}{dx} = 0$ to find critical points, and analyze the sign of $\frac{dy}{dx}$ in each interval to determine increasing and decreasing behavior.

Rewrite the absolute value function as a piecewise function, find the derivative of each piece, determine critical points, and analyze the sign of the derivative in each interval.

Find the derivative, set it equal to zero to find critical points, and analyze the sign of the derivative in each interval, considering the periodic nature of the trigonometric functions.