Equation of tangent line calculator

It evaluates the polynomial to find the contact point, differentiates it to obtain the instantaneous slope, and then calculates the vertical intercept. The result includes both slope-intercept form and point-slope form, so it is useful for checking calculus exercises, preparing worked examples, or adding a predictable symbolic-numeric step to an application.

Enter the polynomial and point clearly

Write the function as an ordinary polynomial in x, such as 3x^2 - 2x + 1, and provide the x-coordinate where the tangent is required. You may include spaces, an optional multiplication sign between a number and x, and either ^ or ** before a nonnegative integer exponent. A leading y= or f(x)= prefix is also accepted. Decimal coefficients and scientific notation are supported, which makes the calculator suitable for both textbook expressions and coefficients obtained from measurements. The expression must remain a polynomial: variables other than x, parentheses, division by x, roots, trigonometric functions, negative exponents, and fractional exponents are rejected instead of being interpreted approximately. This strict input contract prevents an unsupported expression from silently producing a plausible but incorrect line. The x-coordinate must be a finite number. After validation, like powers are combined automatically, so an expression such as x^2 + 2x^2 is treated as 3x^2 before the point and slope are calculated.

How the slope and intercept are derived

For a polynomial f(x), the tangent at x=a passes through the point (a, f(a)) and has slope f'(a). The calculator evaluates the polynomial and its first derivative together using a bounded form of Horner's method. This avoids constructing an expanded derivative string and gives a direct numeric result for both the y-coordinate and the slope. Once m=f'(a) is known, the tangent begins in point-slope form: y-f(a)=m(x-a). Rearranging that identity gives y=mx+b, where b=f(a)-ma. The returned slope and intercept are therefore linked to the displayed equation rather than being separate estimates. For example, a horizontal tangent has slope zero and is formatted simply as y=b. Coefficients are cleaned to stable numeric precision, including removal of negative zero, so equivalent computations produce consistent JSON. If the requested values overflow finite arithmetic, the input is rejected rather than returning Infinity or a damaged equation.

Read and verify the result

The result reports x and y for the point of tangency, slope for the derivative at that point, intercept for the constant term of the line, and equation for the copy-ready slope-intercept form. It also returns a point-slope equation, which preserves the substitution used in the derivation and can be easier to compare with handwritten work. A quick verification is to substitute the returned x into y=mx+b: the result should equal the returned y. You can also differentiate the original polynomial independently and evaluate that derivative at x; it should match slope. These two checks confirm the two defining properties of a tangent line at a regular polynomial point: the line passes through the curve and has the same instantaneous rate of change there. Browser use is free, while automated requests use the published $0.002 base price. Because the algorithm has no network access, randomness, stored state, or clock dependency, the same valid input produces the same structured result in every supported channel.