Practice Tangent Line in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A line that touches a curve at exactly one point and has the same slope as the curve there.

The tangent line is the unique straight line that best approximates the curve at a specific point — same value, same slope.

Showing a random 20 of 50 problems.

Example 1

medium
Use the tangent line of f(x)=xf(x) = \sqrt{x} at x=4x = 4 to approximate 4.1\sqrt{4.1}.

Example 2

easy
Find the slope of the tangent to f(x)=x3f(x) = x^3 at x=2x = 2.

Example 3

easy
Find the slope of the tangent to f(x)=x2−3xf(x) = x^2 - 3x at x=4x = 4.

Example 4

medium
Find all points where f(x)=x3−3xf(x) = x^3 - 3x has a horizontal tangent.

Example 5

easy
Find the equation of the tangent line to h(x)=3x3−xh(x) = 3x^3 - x at x=1x = 1.

Example 6

medium
At what point does f(x)=x2f(x) = x^2 have a tangent line parallel to y=6xy = 6x?

Example 7

easy
Find the equation of the tangent to f(x)=x2−4f(x) = x^2 - 4 at x=1x = 1.

Example 8

medium
Find the tangent to f(x)=x2−3xf(x) = x^2 - 3x at x=2x = 2.

Example 9

easy
Find the point of tangency on f(x)=x3f(x) = x^3 at x=1x = 1.

Example 10

medium
Find the tangent line to f(x)=ln⁡xf(x) = \ln x at x=ex = e.

Example 11

medium
Use the tangent line of f(x)=xf(x) = \sqrt{x} at x=25x = 25 to approximate 26\sqrt{26}.

Example 12

hard
Find the equation of the tangent to the implicit curve x2+y2=25x^2 + y^2 = 25 at the point (3,4)(3, 4).

Example 13

medium
Find the equation of the normal line to f(x)=x2f(x) = x^2 at x=1x = 1.

Example 14

medium
Find the equation of the tangent to f(x)=1xf(x) = \frac{1}{x} at x=2x = 2.

Example 15

hard
Find the tangent line to the curve x2+xy+y2=7x^2 + xy + y^2 = 7 at (1,2)(1, 2).

Example 16

challenge
Find where the tangent to f(x)=x3f(x) = x^3 at x=ax = a also crosses the curve again.

Example 17

hard
At what point on f(x)=exf(x) = e^x is the tangent line parallel to y=e⋅x+7y = e \cdot x + 7?

Example 18

medium
Find the equation of the tangent to f(x)=x3−xf(x) = x^3 - x at x=1x = 1.

Example 19

easy
What is the slope of a horizontal tangent line?

Example 20

easy
Find the slope of the tangent to f(x)=xf(x) = \sqrt{x} at x=4x = 4.
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Related Concepts

SlopeDerivative