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What is the notation for the second derivative?
$f''(x)$, $y''$, $\frac{d^2y}{dx^2}$
What is the notation for the nth derivative?
$f^n(x)$, $\frac{d^ny}{dx^n}$
What is the Power Rule formula?
If $f(x) = ax^n$, then $f'(x) = nax^{n-1}$.
What is the derivative of $\sin(x)$?
$(\sin(x))' = \cos(x)$
What is the derivative of $\cos(x)$?
$(\cos(x))' = -\sin(x)$
What is the Chain Rule formula?
If $f(x) = O(I(x))$, then $f'(x) = O'(I(x)) * I'(x)$.
What is the Product Rule formula?
If $f(x) = L(x) * R(x)$, then $f'(x) = L'(x) * R(x) + L(x) * R'(x)$.
What is the derivative of $\tan(x)$?
$(\tan(x))' = \sec^2(x)$
What is the derivative of $\ln(x)$?
$(\ln(x))' = \frac{1}{x}$
What is the Quotient Rule formula?
If $f(x) = \frac{N(x)}{D(x)}$, then $f'(x) = \frac{D(x)N'(x) - N(x)D'(x)}{(D(x))^2}$.
What is a higher-order derivative?
The derivative of a derivative. It can be the second derivative, third derivative, or any subsequent derivative.
What does the first derivative, $f'(x)$, tell us?
The slope of the function, where the function is increasing or decreasing, and locations of relative minima or maxima.
What does the second derivative, $f''(x)$, tell us?
The concavity of the function and helps us find inflection points (where the concavity changes).
Define inflection point.
A point on a curve where the concavity changes.
What does concavity describe?
The direction in which a curve bends. It can be concave up or concave down.
Define relative minima.
A point where the function's value is less than or equal to the values at all nearby points.
Define relative maxima.
A point where the function's value is greater than or equal to the values at all nearby points.
What is the Power Rule?
A method for differentiating power functions.
What is the Chain Rule?
A method for differentiating composite functions.
What is the Product Rule?
A method for differentiating the product of two functions.
How do you find higher-order derivatives?
To find the *n*th derivative, take the derivative of the *(n-1)*th derivative.
Explain the relationship between the first derivative and increasing/decreasing intervals.
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.
Explain the relationship between the second derivative and concavity.
If $f''(x) > 0$, the function is concave up. If $f''(x) < 0$, the function is concave down.
How do you find inflection points?
Set $f''(x) = 0$ and solve for x. Then, verify that the concavity changes at those x-values.
When do you use the Chain Rule?
When differentiating a composite function (a function within a function).
When do you use the Product Rule?
When differentiating a function that is the product of two other functions.
When do you use the Quotient Rule?
When differentiating a function that is the quotient of two other functions.
What does $f'''(x)$ represent?
The rate of change of the concavity of $f(x)$.
What does it mean if $f'(x) = 0$ and $f''(x) > 0$?
The function has a local minimum at that point.
What does it mean if $f'(x) = 0$ and $f''(x) < 0$?
The function has a local maximum at that point.