# How do you find the slope of a tangent line to the graph of the function #g(x) = 14 − x^2# at (2, 10)?

##### 3 Answers

#### Explanation:

#"the slope of the tangent line "=f'(2)#

#f'(x)=-2x#

#f'(2)=-2(2)=-4larrcolor(blue)"slope of tangent line"#

Part 2 of 2

Using **FIRST PRINCIPLES** with explanation about determining the slope

#### Explanation:

Slope (gradient) is

Suppose we have two points on the graph.

Let the first point be

Let the second point be

Then

This you should have seen before!

In Calculus you start to bring the two points so close together that you can not physically measure the gap between them.

Suppose instead of using

These days people use

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Set the fist point as

Let the next point progress by some minuscule value in

Set

Then the progression point gives us:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

But we need the gradient which is

but change in

We now make

Part 1 of 2

Using shortcut:

#### Explanation:

Given point

Set

If we have

The constant of

So in this case:

The slop

In other words it is the UNIT RATE OF CHANGE.

So at