Find the Tangent Line

Definition:  We know how to find the slope of the tangent line. By taking the derivative at a point, x, we can quite easily find an equation for a line tangent to a curve f(x) at a point x = a.  We will use the point-slope formula for a line, which you may recall from algebra. The equation for a line with slope m through a point (x_1, y_1) is:

tangent2

Example:  Find the equation of a line tangent to the curve f(x) = x^2 + 7x - 2 at the point (1, 6).

Solution:  First, we find the slope of the tangent line by taking the derivative at x = 1: f'(x) = 2x + 7; f'(1) = 9. Then, we use the point-slope equation to create a line equation:

y - 6 = 9(x - 1)

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