Precalculus : Derivatives
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Example Questions
Example Question #5 : Maximum And Minimum Problems
Find the coordinates of the relative maximum point of the function
Let's find the first derivative to locate the relative maxima and minima.
Now we set it equal to zero to find the x values of these critical points.
So the equation is 0 where x is -2, 0, or 5. Now let's find the second derivative so that we know which of these locations are maxima and which are minima.
So the function has a relative maximum at x=-5.
So the function has a relative minimum at x=0.
So the function has a relative maximum at x=2.
Thus there is only one relative minimum in this function, and it occurs at x=0. We need to plug this into the original function to find the y-coordinate of the point.
So our point is (0,8).
Example Question #1 : Maximum And Minimum Problems
Find the 
There are no critical points.
We need to find out where the first derivative is equal to zero to find the locations of the possible maxima and minima.
Example Question #1 : Maximum And Minimum Problems
Without solving the problem, determine whether the function 
One minimum and one maximum
One relative minimum
No critical points
One relative maximum
One relative minimum
We see that the function is a quadratic, so its graph will be a parabola. Thus, we know it will have only one relative max or min. Since the leading coefficient of 
Example Question #1 : Find The Second Derivative Of A Function
Find the second derivative of the following function:
In order to take any order derivative of a polynomial, all we need to know is how to apply the power rule to a simple term with an exponent:
The formula above tells us that to take the derivative of a term with coefficient 
Now if we take the derivative of the first derivative, we'll get the second derivative of our function:
Example Question #2 : Find The Second Derivative Of A Function
Find the second derivative of the function 
To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient 
Applying this rule to each term in the function, we start by taking the first derivative:
Taking the second derivative:
Example Question #21 : Derivatives
Find the second derivative of the function
To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient
Applying this rule to each term in the function, we start by taking the first derivative:
Finally, we take the second derivative:
Example Question #3 : Find The Second Derivative Of A Function
Find the second derivative of the function
To take the derivative of a function, we'll need to apply the power rule to a term with a coefficient
Applying this rule to each term in the function, we start by taking the first derivative:
Then, taking the second derivative of the function:
Example Question #3 : Find The Second Derivative Of A Function
What is the second derivative of
with respect to
We first apply Power Rule.
First Derivative 
So result is
Anything to a power of 
First Derivative is
Second Derivative 
Any derivative of a constant is
Second Derivative of 
Example Question #3 : Find The Second Derivative Of A Function
Find the second derivative of
with respect to
Use Power Rule to take two derivatives of 
First Derivative:
So result is:
Now we take another derivative:
Second Derivative:
So our result is:
Example Question #5 : Find The Second Derivative Of A Function
What is the second derivative of
with respect to
Apply Power Rule twice.
First Derivative of 
So our result is
Second Derivative of 
So our result is
So the second derivative of 
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