Tangent lines worksheet

The derivative at a point is the slope of the curve there, and a point and a slope are all a line needs. The normal is the perpendicular one, so its slope is the negative reciprocal.



Sigma Prep · sigmaprep.io/worksheets Name Tangent and Normal Lines Date Period Find each line or point. Find the tangent line Tangent to $y = 4 x^{3} + 4 x^{2} + 6 x$ at $x = 0$ Tangent to $y = 4 x^{3} + 5 x - 7$ at $x = 2$ Tangent to $y = - 3 x^{2} - 7 x$ at $x = 3$ Find the normal line Normal to $y = - x^{2} - 4$ at $x = - 2$ Find the equation of the normal line to $y = \sqrt{x + 14}$ at $x = 2$ Normal to $y = x^{2} - x$ at $x = - 1$ Find the equation of the normal line to $y = \frac{- 1}{x^{2} + 2}$ at $x = 2$ Normal to $y = - x^{3} + 7 x^{2} + 3$ at $x = 1$ Find the equation of the normal line to $y = \sqrt{4 x - 11}$ at $x = 3$ Find tangent lines to non-polynomial curves Tangent to $y = - 3 \sin x$ at $x = π$ Tangent to $y = 4 \cos x$ at $x = \frac{π}{2}$ Tangent to $y = - 3 \sqrt{x}$ at $x = 4$ Find where the tangent is parallel to a line Find the $x$ -values on $y = x^{3} - 3 x$ where the tangent is parallel to $y = 9 x - 4$ Find the point on $y = x^{2} + 4 x$ where the tangent is parallel to $y = 4 x - 4$ Find the point on $y = x^{2} - 4 x$ where the tangent is parallel to $y = 8 x - 5$ Find tangent lines using the chain or quotient rule Find the equation of the tangent line to $y = {(x + 7)}^{\frac{3}{2}}$ at $x = - 3$ Find the equation of the tangent line to $y = \frac{3}{x^{2} + 1}$ at $x = 2$ Find the equation of the tangent line to $y = \frac{- 4}{x^{2} + 4}$ at $x = 3$ Find where the tangent is horizontal Find the $x$ -values where $y = 2 x^{3} - 24 x + 8$ has a horizontal tangent. Find the $x$ -values where $y = \sqrt{5 x - 2}$ has a horizontal tangent. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep



Sigma Prep · sigmaprep.io/worksheets Name Tangent and Normal LinesAnswer keyVersion 1 Date Period Find each line or point. Find the tangent line Tangent to $y = 4 x^{3} + 4 x^{2} + 6 x$ at $x = 0$ $y = 6 x$ Tangent to $y = 4 x^{3} + 5 x - 7$ at $x = 2$ $y = 53 x - 71$ Tangent to $y = - 3 x^{2} - 7 x$ at $x = 3$ $y = - 25 x + 27$ Find the normal line Normal to $y = - x^{2} - 4$ at $x = - 2$ $y = - \frac{1}{4} x - \frac{17}{2}$ Find the equation of the normal line to $y = \sqrt{x + 14}$ at $x = 2$ $y = - 8 x + 20$ Normal to $y = x^{2} - x$ at $x = - 1$ $y = \frac{1}{3} x + \frac{7}{3}$ Find the equation of the normal line to $y = \frac{- 1}{x^{2} + 2}$ at $x = 2$ $y = - 9 x + \frac{107}{6}$ Normal to $y = - x^{3} + 7 x^{2} + 3$ at $x = 1$ $y = - \frac{1}{11} x + \frac{100}{11}$ Find the equation of the normal line to $y = \sqrt{4 x - 11}$ at $x = 3$ $y = - \frac{1}{2} x + \frac{5}{2}$ Find tangent lines to non-polynomial curves Tangent to $y = - 3 \sin x$ at $x = π$ $y = 3 x - 3 π$ Tangent to $y = 4 \cos x$ at $x = \frac{π}{2}$ $y = - 4 x + 2 π$ Tangent to $y = - 3 \sqrt{x}$ at $x = 4$ $y = - \frac{3}{4} x - 3$ Find where the tangent is parallel to a line Find the $x$ -values on $y = x^{3} - 3 x$ where the tangent is parallel to $y = 9 x - 4$ $x = - 2, x = 2$ Find the point on $y = x^{2} + 4 x$ where the tangent is parallel to $y = 4 x - 4$ $(0, 0)$ Find the point on $y = x^{2} - 4 x$ where the tangent is parallel to $y = 8 x - 5$ $(6, 12)$ Find tangent lines using the chain or quotient rule Find the equation of the tangent line to $y = {(x + 7)}^{\frac{3}{2}}$ at $x = - 3$ $y = 3 x + 17$ Find the equation of the tangent line to $y = \frac{3}{x^{2} + 1}$ at $x = 2$ $y = - \frac{12}{25} x + \frac{39}{25}$ Find the equation of the tangent line to $y = \frac{- 4}{x^{2} + 4}$ at $x = 3$ $y = \frac{24}{169} x - \frac{124}{169}$ Find where the tangent is horizontal Find the $x$ -values where $y = 2 x^{3} - 24 x + 8$ has a horizontal tangent. $x = - 2$ and $x = 2$ Find the $x$ -values where $y = \sqrt{5 x - 2}$ has a horizontal tangent.None Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

How to do these

Tangent to $y = x^{2}$ at $x = 3$

The derivative is 2x, so the slope is 6.
The point is (3, 9).
y - 9 = 6(x - 3).

The answer is $y = 6 x - 9$ .

Where students go wrong

Using the derivative as the y value. It gives the slope; the point still comes from the original function, and mixing the two puts the line in the wrong place.

Also called equation of a tangent line, normal line or horizontal tangent.

What you can put on this worksheet

Tangent lines: Tangent to $y = x^{2} + 2$ at $x = - 1$ → $y = - 2 x + 1$
Normal lines: Normal to $y = x^{2} + 2$ at $x = - 1$ → $y = \frac{1}{2} x + \frac{7}{2}$
Where the tangent is horizontal: Where is the tangent horizontal? $y = x^{3} + 6 x^{2} + 3$ → $x = - 4, x = 0$
Trig, exponential, log, roots and reciprocals: Tangent to $y = 5 e^{- 2 x}$ at $x = 0$ → $y = - 10 x + 5$
Parallel to a given line: Find the point on $y = 2 x^{2}$ where the tangent is parallel to $y = 4 x + 2$ → $(1, 2)$
Tangents needing the chain or quotient rule: Find the equation of the tangent line to $y = \ln (3 x + 10)$ at $x = - 3$ → $y = 3 x + 9$
Normals needing the chain or quotient rule, and vertical normals: Find the equation of the normal line to $y = \ln (x + 1)$ at $x = 0$ → $y = - x$
Horizontal tangents: none, one, two, or on an interval: Find the $x$ -values where $y = 3 x^{2} - 5 x + 1$ has a horizontal tangent. → $x = \frac{5}{6}$

Questions about these worksheets

Yes. Every tangent lines sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the tangent lines questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 8: tangent lines, normal lines, where the tangent is horizontal, trig, exponential, log, roots and reciprocals, parallel to a given line, tangents needing the chain or quotient rule, normals needing the chain or quotient rule, and vertical normals and horizontal tangents: none, one, two, or on an interval. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the derivative as the y value. It gives the slope; the point still comes from the original function, and mixing the two puts the line in the wrong place.

Tangent lines worksheet

Select your difficulty

Find the tangent line

Find the normal line

Find tangent lines to non-polynomial curves

Find where the tangent is parallel to a line

Find tangent lines using the chain or quotient rule

Find where the tangent is horizontal

Find the tangent line

Find the normal line

Find tangent lines to non-polynomial curves

Find where the tangent is parallel to a line

Find tangent lines using the chain or quotient rule

Find where the tangent is horizontal

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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