Answer:
The tangent line to the given curve at the given point is 
.
Step-by-step explanation:
To find the slope of the tangent line we to compute the derivative of 
and then evaluate it for 
.
Differentiate the equation.
Differentiate both sides.
Sum/Difference rule applied: 
Constant multiple rule applied: 
Applied power rule: 
Simplifying and apply constant rule: 
Simplify.
Evaluate y' for x=4:
is the slope of the tangent line.
Point slope form of a line is:
where 
is the slope and 
is a point on the line.
Insert 9 for and (4,10) for :
The intended form is 
which means we are going need to distribute and solve for 
.
Distribute:
Add 10 on both sides:
The tangent line to the given curve at the given point is .
------------Formal Definition of Derivative----------------
The following limit will give us the derivative of the function 
at (the slope of the tangent line at ):
We are given f(4)=10.
Let's see if we can factor the top so we can cancel a pair of common factors from top and bottom to get rid of the x-4 on bottom:
Let's check this with FOIL:
First: 
Outer: 
Inner: 
Last: 
---------------------------------Add!
So the numerator and the denominator do contain a common factor.
This means we have this so far in the simplifying of the above limit:
Now we get to replace x with 4 since we have no division by 0 to worry about:
2(4)+1=8+1=9.
