Tangents and Normal Lines

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      As you may recall, a line which is tangent to a curve at a point a, must have the same slope as the curve. Therefore, the slope of the tangent is

           m = lim   f(a + h) - f(a)
                  ^{h-->0         h}

Since the slope equation of the tangent line is exactly the same as the derivative definition, an easier way to find the tangent line is to differentiate using the rules on the function f. For example,

Find the slope of a line tangent to the function f(x) = x² + 1.
f '(x) = 2x
The slope of the tangent line for all points on the graph is 2x.

To find the slope and equation of a line tangent to a certain point, you must:

First find the slope of the function by differentiation.
Second plug in the certain point's values for x and y
Finally plug both the slope and point values into a linear equation. Observe the example below.

Find the equation of the tangent line for the function f(x) = x² + 1 at point (3,10).

Find the slope of the function by differentiation
f '(x) = 2x

Plug in the certain point's values Since this function does not have y we don't plug in y yet
f '(3) = 6 {6 is now the slope of the point 3,10}

Plug both slope and point values into a linear equation
(y - y₁) = m(x - x₁) {this is the linear equation}
(y - 10) = 6(x - 3)  {Which can be simplified as below}
y = 6x -8

Just as we can find the slope and equation of a tangent line for a function, we can also do the same for a normal line. However, the normal line has two differences from the tangent line.

1. The slope of a normal line is perpendicular to the slope of the tangent line. Or in other words, the negative inverse of the tangent line.

2. The normal line is only defined if x does not = 0.

As a result, to find the slope and equation of the normal line, follow the steps above and convert the slope of the tangent line to the slope of the normal line.