Tangent Line Calculator at a Point
This tangent line calculator takes a function and an x-value and gives you the slope of the
function at that point and the equation of the tangent line as y = mx + b. It also shows the
derivative, simplified and typeset like a textbook. The Function Slope Calculator works in radians or
degrees, shows the final answer only (no step-by-step working), and runs in your browser with no sign-up.
How to find the tangent line at a point
- Type the function into Function, for example
x^2 + 2x. - Type the point into x-value (optional), for example
3. - Read the Slope at x = 3 card for the slope, and the Tangent line card for the equation.
x-value: 3
Derivative: 2x + 2
Slope at x = 3: 8
Tangent line: y = 8x − 9
The results update about a tenth of a second after you stop typing. While an expression is half-typed, the calculator keeps the last result and dims it after a second. Leave the x-value empty to see only the derivative.
How the slope and the tangent line are calculated
The slope of a function at a point is the value of its derivative there, m = f'(a). The tangent line passes through the point (a, f(a)) with that slope, so its intercept is b = f(a) − m·a.
For x^2 + 2x at 3: the derivative is 2x + 2, so m = 8. The function value is
f(3) = 15, so b = 15 − 8·3 = −9, and the tangent line is y = 8x − 9. The line is written the
way a textbook would write it: no 1x, no + 0, and just y = b when the
slope is 0.
Slope of a function at a point: more examples
ln(x) at 1: slope 1, tangent line y = x − 1
e^x at 0: slope 1, tangent line y = x + 1
The x-value can be a number or a constant expression such as pi/2. Exact fractions and symbols
stay exact in the derivative, such as 1/(3x^(2/3)) for cbrt(x); type a decimal
such as 0.5x^2 and the derivative keeps decimals. The slope and the tangent line are shown to
8 significant digits.
How to type a function
Write powers as x^2, products as 3x or 2*x, and functions like
sin(x), sqrt(x), e^x and ln(x). Put brackets after a
function name, as in sin(2x).
| You type | It means |
|---|---|
x^2, x^(1/3) | Powers, including fractional and negative powers |
3x, 2*x, x(x+1) | Multiplication, written out or implied |
pi, e | The constants pi and Euler's number |
ln(x), log(x), log10(x), log2(x) | Logarithms |
sqrt(x), cbrt(x), abs(x), exp(x) | Roots, absolute value, e to the power x |
Trigonometric functions (sin, cos, tan and the rest, with their
inverses) and hyperbolic functions (sinh, cosh, tanh and the rest) are
supported. floor, ceil, round, gamma and
erf are not.
Radians or degrees
Open Settings and choose Radians (the default) or Degrees
under Angles in trigonometry. In degrees, sin(x) means the sine of x degrees,
inverse trigonometric functions return degrees, and the x-value is read in degrees too. Because one degree
is pi/180 radians, the derivative carries a factor of pi/180, and a note under the result says so.
x-value: 0
Slope at x = 0: 0.017453293 (that is pi/180)
Tangent line: y = 0.017453293x
When there is no slope at a point
Where a function has no single slope, the card says undefined at this point (no single slope here).
That happens for sqrt(x) at 0 (the tangent is vertical), abs(x) at 0 (a corner),
1/x at 0, ln(x) at a negative number and tan(x) at
pi/2. If the function uses another letter, such as a*x, the card says
"Can't evaluate: a has no value" until you replace the letter with a number.
What this tangent line calculator does not do
It finds the first derivative of a function of x and the tangent line at one point, and shows the result only. It does not give steps, and it does not do implicit, parametric or polar curves, partial or higher-order derivatives, or differentiating with respect to a letter other than x. For that, or for a derivative that is not tied to a point, use the Derivative Calculator.
Frequently asked questions
How do I find the equation of a tangent line at a point?
Type the function into the Function field and the point into the x-value (optional) field. The Tangent line card then shows the line as y = mx + b. For x^2 + 2x at x = 3 the slope is 8 and the card shows y = 8x − 9. By hand, the slope m is the derivative at the point and b is f(a) − m·a, where a is the x-value.
How do I find the slope of a function at a point?
Enter the function and the x-value. The Slope at x card gives the derivative's value there, to 8 significant digits. The x-value can be a number or a constant expression such as pi/2. For sqrt(x) at 4 the slope is 0.25.
What is the derivative of sin x in degrees?
If x is measured in degrees, the derivative of sin(x) is (π/180)·cos(x), because one degree is π/180 radians. Open Settings, set Angles in trigonometry to Degrees, and the calculator applies that factor for you: sin(x) at x = 0 gives a slope of 0.017453293.
Does the tangent line calculator show steps?
No. It shows the function as read, the simplified derivative, the slope at the point and the tangent line, but not the working. The worked example on this page shows how the slope and intercept are found, so you can check each number by hand.
Can it find a tangent line to a parametric, polar or implicit curve?
No. It handles a function written as y in terms of x, and it differentiates with respect to x only. Parametric, polar and implicit curves, partial derivatives and second derivatives are not supported.
Can I use it for tangent line approximation?
Yes, for the line itself. The tangent line y = mx + b is the linear approximation of the function near your x-value, so you can read it off and substitute a nearby x yourself. The calculator does not evaluate the approximation at a second point or give the error.
What does "undefined at this point (no single slope here)" mean?
The function has no single tangent slope at that x-value. That happens at a corner such as abs(x) at 0, at a vertical tangent such as sqrt(x) at 0, where the function blows up like 1/x at 0, outside its domain like ln(x) at a negative number, and at tan(x) at pi/2.
Does the calculator send my function to a server?
No. The function is read and differentiated in your browser, so nothing you type is sent anywhere, and there is no account or sign-up.
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