Normal Line Calculator
Enter a function \( y = f(x) \) and the x-value where you want the normal. We’ll differentiate the function, compute the tangent slope, and then give you the normal line equation (and tangent line too).
Use x only. Examples: x^2, sin(x), e^x, x^3 - 2x + 1.
We’ll evaluate f(x) there.
Formulas used
1. Tangent slope at \(x = x_0\): \(m_t = f'(x_0)\)
2. Tangent line: \(y - y_0 = m_t (x - x_0)\)
3. Normal slope: \(m_n = -\frac{1}{m_t}\) (if \(m_t \neq 0\))
4. Normal line: \(y - y_0 = m_n (x - x_0)\)
5. Horizontal tangent \(m_t = 0\) → normal is vertical: \(x = x_0\)
6. Vertical tangent (rare; infinite slope) → normal is horizontal: \(y = y_0\)
How to use this normal line calculator
- Type your function in standard math notation.
- Enter the x-value where you want the normal.
- Click Calculate to see tangent slope, tangent line, and normal line.
- Copy the equation form you need for homework or plotting.
Unlike some competing tools, this one also shows the tangent line alongside the normal and handles the typical “horizontal tangent” and “vertical tangent” edge cases explicitly.
FAQs
Can I use trig and exponentials?
Yes. You can enter sin(x), cos(x), tan(x), exp(x), log(x), etc.
What if the derivative doesn’t exist?
We’ll try to differentiate symbolically. If the point is not in the domain or the derivative is undefined, we’ll tell you so.
Is this the same as a tangent line calculator?
Very close — a normal line is just the line perpendicular to the tangent. That’s why we show both.
Formula (LaTeX) + variables + units
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1. Tangent slope at \(x = x_0\): \(m_t = f'(x_0)\) 2. Tangent line: \(y - y_0 = m_t (x - x_0)\) 3. Normal slope: \(m_n = -\frac{1}{m_t}\) (if \(m_t \neq 0\)) 4. Normal line: \(y - y_0 = m_n (x - x_0)\) 5. Horizontal tangent \(m_t = 0\) → normal is vertical: \(x = x_0\) 6. Vertical tangent (rare; infinite slope) → normal is horizontal: \(y = y_0\)
- T = property tax (annual or monthly depending on input) (currency)
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.