Please show how you did each?
1 Answer
See below. Each part is bolded.
Explanation:
1. The limit definition for solving a derivative is as follows:
So, let's start by plugging the equation in:
Simplify:
Next, factor out
The
Finally, apply the limit using substitution:
2. Plug the equation into the alternate limit definition:
Simplify:
The numerator factors:
Simplify:
Finally, apply the limit using substitution:
3. The derivative rules we will use here will be the power rule, the sum rule, and the constant rule:
This is the power rule:
This is the sum rule:
This is the constant rule, where
Let's start with the sum rule:
First, we will equate
Using the sum rule, we will differentiate each term individually:
Next, use the power rule to take the derivative of the first two terms:
By the constant rule, the derivative of any constant is 0:
Finally, simplify:
4. The derivative of a function at a certain point gives the slope of the line tangent to that point.
Horizontal lines have a slope of zero, so we can set our derivative equal to zero, to get the x value for that point:
Now, plug
Simplify:
So, we know that the function has a tangent line with a slope of zero (horizontal) at
5. We know from Part 2 that the slope of the tangent line at
Now, plug
Simplify:
Finally, plug this information into the point-slope formula:
This can also be written in slope intercept form if