$f''(x)$, $y''$, $\frac{d^2y}{dx^2}$

$f^n(x)$, $\frac{d^ny}{dx^n}$

If $f(x) = ax^n$, then $f'(x) = nax^{n-1}$.

$(\sin(x))' = \cos(x)$

$(\cos(x))' = -\sin(x)$

If $f(x) = O(I(x))$, then $f'(x) = O'(I(x)) * I'(x)$.

If $f(x) = L(x) * R(x)$, then $f'(x) = L'(x) * R(x) + L(x) * R'(x)$.

$(\tan(x))' = \sec^2(x)$

$(\ln(x))' = \frac{1}{x}$

If $f(x) = \frac{N(x)}{D(x)}$, then $f'(x) = \frac{D(x)N'(x) - N(x)D'(x)}{(D(x))^2}$.

To find the nth derivative, take the derivative of the *(n-1)*th derivative.

If $f'(x) > 0$, the function is increasing. If $f'(x) < 0$, the function is decreasing. If $f'(x) = 0$, there may be a local max or min.

If $f''(x) > 0$, the function is concave up. If $f''(x) < 0$, the function is concave down.

Set $f''(x) = 0$ and solve for x. Then, verify that the concavity changes at those x-values.

When differentiating a composite function (a function within a function).

When differentiating a function that is the product of two other functions.

When differentiating a function that is the quotient of two other functions.

The rate of change of the concavity of $f(x)$.

The function has a local minimum at that point.

The function has a local maximum at that point.

The derivative of a derivative. It can be the second derivative, third derivative, or any subsequent derivative.

The slope of the function, where the function is increasing or decreasing, and locations of relative minima or maxima.

The concavity of the function and helps us find inflection points (where the concavity changes).

A point on a curve where the concavity changes.

The direction in which a curve bends. It can be concave up or concave down.

A point where the function's value is less than or equal to the values at all nearby points.

A point where the function's value is greater than or equal to the values at all nearby points.

A method for differentiating power functions.

A method for differentiating composite functions.

A method for differentiating the product of two functions.