Find dy/dx by implicit differentiation

How it works

Write the equation as H(x, y) = 0, where H is the left side minus the right side (with no = sign, the page reads the typed expression as H = 0). Then

dy/dx = −H_x / H_y

where H_x = ∂H/∂x (y held fixed) and H_y = ∂H/∂y (x held fixed).

The page differentiates H itself, with the sum, product, quotient, power and chain rules (for u^c with a constant c, c u^(c−1) u′; for u^v otherwise, u^v (v′ ln u + v u′/u)) and the derivatives of sin, cos, tan, exp, ln, sqrt, asin, acos, atan, sinh, cosh and abs. Zeros and ones are dropped as it goes, and numbers in a product are multiplied together. Any other function gives no answer.

Writing the answer. Let g be the largest whole number that divides the number in front of every term of H_x and of H_y (a term with no number in front, or with a number that is not whole, counts as 1; g is 1 when there is no larger one). Both are divided by g. The minus sign goes into the top: each term of H_x changes sign. dy/dx is the top over H_y (or the top alone when H_y is 1), written by the page: for x³ + y³ = 6xy, H_x = 3x² − 6y and H_y = 3y² − 6x, g = 3, and dy/dx = (2y − x^2)/(y^2 − 2x). The answer is not simplified further (−y 2^(x y) ln(2)/(x 2^(x y) ln(2)) keeps its common factors).

The answer is checked. At the test points (x, y) = (0.37, 1.21), (1.73, −0.61), (−2.17, 0.83), (2.9, 3.3), (−0.53, −1.9) and (0.81, 2.47), the formula must agree with −(H(x + h, y) − H(x − h, y))/(H(x, y + h) − H(x, y − h)), h = 10⁻⁵, to 1 part in 10⁵ of max(1, |formula|), at every point where both are real numbers, and there must be at least 3 such points. Otherwise the page says "No verified answer".

The slope at a point. If you give a point "x, y", it must be on the curve: |H(x, y)| at most 10⁻⁹ × max(1, |left side at the point|). The slope is dy/dx there, to 10 significant figures (rounded half up); it is left out when dy/dx is not a real number there (a vertical tangent).

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• The equation uses x and y, with one = (or none). Numbers can have decimals; + − * / and ^; brackets; a number or bracket next to a letter multiplies, and a space between letters multiplies (x y, 6x y).
• Constants pi and e; functions sin, cos, tan, exp, ln (log), sqrt, asin, acos, atan (arcsin, arccos, arctan), sinh, cosh, abs (|x|). Angles are in radians.
• The point is two numbers or constant expressions separated by a comma: 3, -4 or 3/2, 3/2.

What gets no answer

• An equation without y (dy/dx has no meaning), with more than one =, or with a function this page does not differentiate.
• A formula that fails its check or can be checked at fewer than 3 test points.
• A point that is not on the curve (a message next to the point).

Worked examples by hand

x² + y² = 25, and the slope at (3, −4) (OpenStax Calculus Volume 1, section 3.8, Examples 3.68 and 3.71). H = x² + y² − 25, H_x = 2x, H_y = 2y, g = 2: dy/dx = −x/y. At (3, −4): −3/(−4) = 0.75.

x³ sin(y) + y = 4x + 3 (OpenStax Calculus Volume 1, section 3.8, Example 3.69). H = x³ sin(y) + y − 4x − 3, H_x = 3x² sin(y) − 4, H_y = x³ cos(y) + 1. dy/dx = (4 − 3x^2 sin(y))/(x^3 cos(y) + 1).

y³ + x³ − 3xy = 0, and the slope at (3/2, 3/2) (OpenStax Calculus Volume 1, section 3.8, Example 3.72). H_x = 3x² − 3y, H_y = 3y² − 3x, g = 3: dy/dx = (y − x^2)/(y^2 − x), the book's (3y − 3x²)/(3y² − 3x). At (3/2, 3/2): (3/2 − 9/4)/(9/4 − 3/2) = −1.