Normal Line Calculator
Work out normal line instantly with clear inputs, formula shown and shareable results.
About Normal Line Calculator
The normal is perpendicular to the tangent at the point of contact, so its slope is the negative reciprocal −1/f'(a). Where the tangent is horizontal the normal becomes a vertical line.
Formula
Normal line
y = f(a) − (1/f'(a))(x − a)
Worked example
With the default values below, Normal Line Calculator returns Normal line of y = -0.166667x + 9.5. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Functionone of a·xᵖ, a·sin(kx), a·cos(kx), a·e^(kx), a·ln(kx), a·√(kx)
- a·xᵖ
- Coefficient a
- 1
- Power p or inner coefficient k
- 2
- Point x
- 3
Results
- Normal line
- y = -0.166667x + 9.5
- Normal slope −1/f'(x)
- -0.16666667
- Tangent slope f'(x)
- 6.00000000
- f(x) at the point
- 9.00000000
Step by step
- 1Values used
- 2Normal line
- 3Normal line= y = -0.166667x + 9.5
- 4Normal slope −1/f'(x)= -0.16666667
- 5Tangent slope f'(x)= 6.00000000
- 6f(x) at the point= 9.00000000
Assumptions and limits
- The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Normal to y = x² at x = 3?
The tangent slope is 6, so the normal slope is −1/6 and the line is y = −x/6 + 9.5.
What if the derivative is zero?
The tangent is horizontal so the normal is the vertical line x = a, which has no gradient.