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Equation of the tangent

The tangent is the straight line that touches the graph at one point and has the same slope as the graph there. The slope is $f'(x_0)$ , and the line passes through the point $(x_0, f(x_0))$ . The point-slope form gives the equation directly.

y - f(x_0) = f'(x_0)(x - x_0)

point-slope form of the tangent

Symbols

$x_0$	x value of the point of contact
$f'(x_0)$	slope at the point

Example

$f(x) = x^2$ at $x_0 = 3$ : $f(3) = 9$ and $f'(3) = 2\cdot 3 = 6$ . Then $y = 6x - 9$ , since $9 - 6\cdot 3 = -9$ .

Check that the tangent passes through the point:

6cdot 3 - 9 = 9

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Part of Mathematics 1T: Differentiation and rate of change.