Math & Conversions Tangent Line Calculator 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. Equation: $f(x)$ and Point $x_0$ Function $f(x)$ Point $x_0$ Calculate Tangent Line Tangent Line Equation ($y = mx + b$) Slope ($m = f'(x_0)$) Point of Tangency $(x_0, y_0)$ Step-by-Step Calculation 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 $x_0$: $$m = f'(x_0)$$ 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: $$y_0 = f(x_0)$$ 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) What is the relationship between the derivative and the tangent line? The derivative of a function $f(x)$ at a specific point $x_0$ is mathematically equal to the slope of the tangent line to the graph of $f(x)$ at that point. $m = f'(x_0)$. What is the formula for the tangent line? The most fundamental formula is the point-slope form: $y - y_0 = m(x - x_0)$, where $m = f'(x_0)$ and $y_0 = f(x_0)$. What is the normal line? The normal line is the line that is perpendicular to the tangent line at the point of tangency. Its slope is the negative reciprocal of the tangent line's slope: $m_{normal} = -1

Subcategories in Math & Conversions Tangent Line Calculator 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. Equation: $f(x)$ and Point $x_0$ Function $f(x)$ Point $x_0$ Calculate Tangent Line Tangent Line Equation ($y = mx + b$) Slope ($m = f'(x_0)$) Point of Tangency $(x_0, y_0)$ Step-by-Step Calculation 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 $x_0$: $$m = f'(x_0)$$ 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: $$y_0 = f(x_0)$$ 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) What is the relationship between the derivative and the tangent line? The derivative of a function $f(x)$ at a specific point $x_0$ is mathematically equal to the slope of the tangent line to the graph of $f(x)$ at that point. $m = f'(x_0)$. What is the formula for the tangent line? The most fundamental formula is the point-slope form: $y - y_0 = m(x - x_0)$, where $m = f'(x_0)$ and $y_0 = f(x_0)$. What is the normal line? The normal line is the line that is perpendicular to the tangent line at the point of tangency. Its slope is the negative reciprocal of the tangent line's slope: $m_{normal} = -1.