Tangent Line Calculator

Find the equation of the tangent line to the function $f(x)$ at the specified point $x_0$. The tangent line is a linear approximation of the curve at that exact point.

The Three Essential Steps to Finding the Tangent Line

The equation of the tangent line is found by applying the point-slope formula, $y - y_0 = m(x - x_0)$. This requires three key components derived from calculus:

1. Find the Slope ($m$)

The slope is the value of the first derivative of the function $f'(x)$ evaluated at the point $x_0$:

2. Find the Point of Tangency ($x_0, y_0$)

The $y$-coordinate ($y_0$) is found by substituting $x_0$ into the original function:

3. Construct the Equation

Substitute $m$, $x_0$, and $y_0$ into the point-slope form and algebraically rearrange it into the common slope-intercept form, $y = mx + b$.

Tangent Line vs. Normal Line

The **Normal Line** is perpendicular to the tangent line at the point of tangency. This relationship is defined by their slopes:

• If the slope of the tangent line is $m$, the slope of the normal line ($m_n$) is the negative reciprocal: $$m_n = -\frac{1}{m}$$

The normal line equation is often used in optics (reflection/refraction) and fluid dynamics.

Frequently Asked Questions (FAQ)

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m = f'(x_0)

y_0 = f(x_0)

m_n = -\frac{1}{m}

y - y_0 = m(x - x_0)

m_{\text{normal}} = -\frac{1}{m}