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How do you find the second derivative of $f(x) = x^4 + 3x^2 - 2x$?
1. Find the first derivative: $f'(x) = 4x^3 + 6x - 2$. 2. Find the derivative of $f'(x)$: $f''(x) = 12x^2 + 6$.
How do you find the second derivative of $f(x) = \sin(3x)$?
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: $f''(x) = -9\sin(3x)$.
How do you find the second derivative of $f(x) = x\cos(x)$?
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: $f''(x) = -\sin(x) - (\sin(x) + x\cos(x)) = -2\sin(x) - x\cos(x)$.
How do you find the second derivative of $f(x) = \frac{x}{x+1}$?
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: $f''(x) = -2(x+1)^{-3} = \frac{-2}{(x+1)^3}$.
How do you find the second derivative of $f(x) = \ln(x^3)$?
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 $f'(x)$: $f''(x) = -\frac{3}{x^2}$.
How to find intervals where $f(x)$ is increasing/decreasing?
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).
How to find intervals where $f(x)$ is concave up/down?
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).
How to find the x-coordinates of inflection points?
1. Find $f''(x)$. 2. Set $f''(x) = 0$ and solve for x. 3. Verify that the concavity changes at each x-value.
How to find the second derivative of $f(x) = e^{2x}$?
1. Find the first derivative using the Chain Rule: $f'(x) = 2e^{2x}$. 2. Find the derivative of $f'(x)$ using the Chain Rule: $f''(x) = 4e^{2x}$.
How do you find the third derivative of $f(x) = 5x^4 - 3x^2 + 7$?
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: $f'''(x) = 120x$.
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.