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

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

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.

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: $f'''(x) = 120x$.