--- title: "Tangents and Normals to a Curve" description: "Find the equation of the tangent at a point on a curve from the derivative there, and the normal from the perpendicular gradient." canonical: https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp source: https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Tangents and Normals to a Curve Find the equation of the tangent at a point on a curve from the derivative there, and the normal from the perpendicular gradient. Subject: Mathematics · Area: Calculus & Analysis · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp ## Ready when they can - Find a point and tangent gradient on a curve, then construct the tangent equation - Construct a normal; check the gradient product when both gradients are finite, and handle horizontal tangents with vertical normals - Find axis crossings of tangent and normal lines from their equations ## Lesson: Tangents and Normals to a Curve A derivative describes a rate of change, with units of output per unit of input. For y against x, a finite dy/dx at a point is the tangent's gradient. A point and a gradient fix a nonvertical line: the curve supplies the point and its derivative supplies the gradient. In the Ridgeway ramp's straight-line model, a board continues along the tangent from the lip. The crew uses its equation to find where the line reaches the ground. Put x into the curve's rule to get y. Differentiate and put the same x into dy/dx to get finite gradient m. Use y-y₁=m(x-x₁). If a tangent gradient is supplied, use it directly. If asked where a curve has a chosen gradient, set dy/dx equal to that value and solve for x. A normal passes through the same point at right angles to the tangent. For finite m≠0, its gradient is -1/m. For m=0, its equation is x=x₁ and its gradient is undefined. *(drawing: Two different rules feed the two slots. Mixing them up is the usual slip.)* **Example.** For y=x²-3x at x=2, the curve gives y=4-6=-2, so the point is (2,-2). Differentiate: dy/dx=2x-3, giving m=1 there. Thus y+2=1(x-2), or y=x-4. Check both the point and gradient: x=2 gives y=-2, and the line's gradient is 1. For reciprocal or root curves, first rewrite the function as powers, then differentiate and use the same point-gradient method. **Example.** A normal meets the tangent at right angles through the same point. With finite tangent gradient m≠0, its gradient is -1/m and the product is -1. At (2,-2), m=1 gives y+2=-(x-2), so y=-x. For a fractional example, y-8=-(1/7)(x-2) becomes x+7y-58=0 after multiplying by 7. Opposite signs alone do not prove perpendicularity: 7 times -7 is -49, not -1. If m=0, the normal is vertical, x=x₁, with no gradient. *(drawing: The product test needs two finite gradients. A horizontal tangent instead pairs with a vertical normal.)* **Example.** For an axis crossing, first build the line. Get (x₁,y₁) from the curve and finite tangent gradient m from dy/dx. Use y-y₁=m(x-x₁) for the tangent, or y-y₁=-(1/m)(x-x₁) for the normal when m≠0. Then set y=0 for an x-axis crossing or x=0 for a y-axis crossing and solve. The tangent y=x-4 meets the axes at (4,0) and (0,-4). These are coordinate crossings, not a claim about a board moving downwards. **Recap.** Point from curve; m from dy/dx. Use y-y₁=m(x-x₁). For finite m≠0, normal gradient -1/m; product -1. For m=0, normal x=x₁. Axis crossings: y=0 for x-axis, x=0 for y-axis. ## Practice 30 questions on this page, each with its working shown. ## Needs first - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka) - [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3) ## Optional connections - [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW) ## Opens up - [Tangent Planes and Linear Approximation](https://lightmysky.com/learn/mathematics/tangent-planes-and-linear-approximation-mt_EYMcjChX_m) - [The Newton-Raphson Method](https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd) - [Increasing and Decreasing Functions](https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ)