What is a tangent line?

A tangent line to a curve at a point is the straight line that just touches the curve at that point and has the same slope as the curve at that instant. The slope of the tangent is the derivative of the function evaluated at the point. Tangent lines are used to approximate functions locally, to find the direction of motion, and as the starting point for linear approximations.

How do you find the equation of a tangent line?

To find the tangent at x = a: first evaluate f(a) to get the y coordinate of the point. Then find the derivative f'(a), which is the slope of the tangent. Finally, use the point-slope form y minus f(a) equals f'(a) times (x minus a), rearranging to y = f'(a) times x plus (f(a) minus f'(a) times a).

What is the normal line?

The normal line at a point is perpendicular to the tangent line at that same point. If the tangent has slope m, the normal has slope -1/m (the negative reciprocal). The normal line equation uses the same point (a, f(a)) but with the perpendicular slope. If the tangent is horizontal (slope 0), the normal is a vertical line x = a.

Tangent Line Calculator

The tangent line to a curve at a point is the straight line that best approximates the curve right at that point. It touches the curve without crossing it (for smooth functions) and has exactly the same slope as the curve at the point of contact. That slope is the derivative of the function, and finding it is the central task of differential calculus. Tangent lines matter practically because they let you approximate complex function values using simple linear arithmetic, they define the direction of motion for a moving object, and they form the basis of Newton's method for finding roots. This calculator takes any function of x written in standard notation, such as x^2, and an x value at which to find the tangent. It returns the slope of the tangent (which equals the derivative at that point), the complete tangent line equation in y = mx + c form, the coordinates of the point of tangency, and the normal line equation for the perpendicular line at the same point. Slope is computed by central difference numerical differentiation with h = 1e-7, giving highly accurate results for smooth functions. Use standard operators +, -, *, / and ^ for powers, plus functions like sin, cos, tan, exp, log, sqrt, and abs. The default example uses f(x) = x^2 at x = 2, giving slope = 4 and tangent y = 4x - 4.

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Function f(x)

x value

4.00

slope of tangent (derivative at x)

Tangent liney = 4x - 4

Point (x, f(x))(2.00, 4.00)

Normal liney = -0.25x + 4.50

Slope uses central difference numerical differentiation. Use ^ for powers and * for multiplication.

How it works

The derivative (slope of the tangent) is estimated by central difference: f'(x) is approximately (f(x+h) - f(x-h)) / (2h) with h = 1e-7. The function value f(x) is the y coordinate of the point of tangency. The tangent line equation uses point-slope form: y = slope * x + (f(x) - slope * x), giving the y-intercept c. The normal line is perpendicular with slope = -1/slope (or described as a vertical line if the tangent is horizontal). Positive c is shown as y = mx + c; negative c is shown as y = mx - |c|.

Worked example

For f(x) = x^2 at x = 2: the function value is f(2) = 4, so the point of tangency is (2.00, 4.00). The derivative is f'(x) = 2x, so f'(2) = 4. The tangent line equation is y = 4x + (4 - 4 * 2) = y = 4x - 4. The normal slope is -1/4 = -0.25 and the normal line is y = -0.25x + (4 + 0.25 * 2) = y = -0.25x + 4.50. These match the defaults pre-filled above.

Related calculators

Average Rate of Change Calculator: find the slope of the secant line between two points.
Critical Points Calculator: find where the tangent slope equals zero.
Partial Derivative Calculator: extend differentiation to functions of two variables.
Area Under the Curve Calculator: integrate any function over an interval.

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