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

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

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

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.

1. Find the first derivative: $f'(x) = 4x^3 + 6x - 2$. 2. Find the derivative of $f'(x)$: $f''(x) = 12x^2 + 6$.

1. Find the first derivative using the Chain Rule: $f'(x) = 3\cos(3x)$. 2. Find the derivative of $f'(x)$ using the Chain Rule: $f''(x) = -9\sin(3x)$.

1. Find the first derivative using the Product Rule: $f'(x) = \cos(x) - x\sin(x)$. 2. Find the derivative of $f'(x)$ using the Product Rule: $f''(x) = -\sin(x) - (\sin(x) + x\cos(x)) = -2\sin(x) - x\cos(x)$.

1. Find the first derivative using the Quotient Rule: $f'(x) = \frac{(x+1)(1) - x(1)}{(x+1)^2} = \frac{1}{(x+1)^2}$. 2. Rewrite $f'(x)$ as $(x+1)^{-2}$. 3. Find the derivative of $f'(x)$ using the Chain Rule: $f''(x) = -2(x+1)^{-3} = \frac{-2}{(x+1)^3}$.

1. Simplify $f(x)$ using logarithm properties: $f(x) = 3\ln(x)$. 2. Find the first derivative: $f'(x) = \frac{3}{x}$. 3. Find the derivative of $f'(x)$: $f''(x) = -\frac{3}{x^2}$.

1. Find $f'(x)$. 2. Set $f'(x)=0$ and find critical points. 3. Create a sign chart for $f'(x)$. 4. Determine intervals where $f'(x)>0$ (increasing) and $f'(x)<0$ (decreasing).

1. Find $f''(x)$. 2. Set $f''(x)=0$ and find possible inflection points. 3. Create a sign chart for $f''(x)$. 4. Determine intervals where $f''(x)>0$ (concave up) and $f''(x)<0$ (concave down).

1. Find $f''(x)$. 2. Set $f''(x) = 0$ and solve for x. 3. Verify that the concavity changes at each x-value.

1. Find the first derivative using the Chain Rule: $f'(x) = 2e^{2x}$. 2. Find the derivative of $f'(x)$ using the Chain Rule: $f''(x) = 4e^{2x}$.

1. Find the first derivative: $f'(x) = 20x^3 - 6x$. 2. Find the second derivative: $f''(x) = 60x^2 - 6$. 3. Find the third derivative: $f'''(x) = 120x$.