Tangent & Normal Line Equation Finder | Calculus Line Solver

TANGENT & NORMAL LINE FINDER

Calculus Slope & Line Equation Solver

Differential Calculus

Select Function Preset: Parabola: f(x) = x² - 4x + 3 at x = 3 Cubic Polynomial: f(x) = x³ - 3x + 2 at x = 2 Trigonometric: f(x) = sin(x) at x = π/3 Exponential: f(x) = e^x at x = 0 Rational Function: f(x) = 1 / x at x = 2

Function $f(x)$:

Evaluation Point $x = c$:

Tangent & Normal Analysis Output

Point on Curve $(c, f(c))$ (3.00, 0.00)

Tangent Slope $m_T = f'(c)$ 2.00000

Tangent Line Equation y = 2.000x - 6.000

Normal Line Equation y = -0.500x + 1.500

Step-by-Step Calculus Derivation

Tangent and Normal Line Equations: Differential Calculus Guide

The tangent line to a curve $y = f(x)$ at a point $x = c$ is the linear approximation that best fits the curve at that exact coordinate, matching both position and instantaneous rate of change. The normal line is perpendicular to the tangent line at the exact same point of contact.

1. Mathematical Formulas

Line Type	Slope Expression	Point-Slope Equation
Tangent Line	$$m_T = f'(c)$$	$$y - f(c) = f'(c)(x - c)$$
Normal Line	$$m_N = -\frac{1}{f'(c)} \quad (f'(c) \neq 0)$$	$$y - f(c) = -\frac{1}{f'(c)}(x - c)$$

2. Step-by-Step Solution Workflow

1. Find the Point: Evaluate $y_0 = f(c)$ to obtain coordinates $(c, y_0)$.
2. Compute the Derivative: Differentiate $f(x)$ symbolically to get $f'(x)$.
3. Calculate Tangent Slope: Substitute $x = c$ into $f'(x)$ to find $m_T = f'(c)$.
4. Build Tangent Line: Apply point-slope form $y - y_0 = m_T(x - c)$ and simplify to $y = m_T x + b_T$.
5. Build Normal Line: Find $m_N = -\frac{1}{m_T}$, then apply $y - y_0 = m_N(x - c)$ to get $y = m_N x + b_N$.

Frequently Asked Questions

What if $f'(c) = 0$?

When $f'(c) = 0$, the tangent line is horizontal ($y = f(c)$). The normal line is perpendicular and thus vertical, represented by $x = c$.

What if $f'(c)$ is undefined or infinite?

If $f'(c) \to \infty$, the tangent line is vertical ($x = c$), while the normal line is horizontal ($y = f(c)$).

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