Math

Slope of a Curve Calculator

Find the Slope of Any Curve at a Point

Unlike straight lines that have a constant slope, curves have a slope that changes from point to point. The slope of a curve at any specific point is given by the derivative evaluated at that point. Enter your function and a point to find the exact slope.

Derivative Calculator Tips

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Try an Example

Pick a scenario to see how the calculator works, then adjust the values

Power Rule

Differentiate a polynomial using the power rule.

Key values: f(x) = x^3 - 2x^2 + x · 1st derivative

Chain Rule

Differentiate a composite function with sin(x^2).

Key values: f(x) = sin(x^2) · Chain rule applied

Product Rule with Tangent

Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.

Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0

Second Derivative

Compute the second derivative to analyze concavity.

Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative

Documentation

This calculator is also known as Slope of a Curve Calculator.

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Slope of a Curve vs Slope of a Line

A straight line has the same slope everywhere. A curve's slope changes depending on where you are on the curve. The derivative gives you a formula for the slope at every point: f'(x) is the slope of the curve y = f(x) at the point x.

Examples

Slope of y = x^3 at x = 1

Find the slope of the curve y = x^3 at the point x = 1.

f'(x) = 3x^2. At x = 1: slope = 3(1)^2 = 3.

Key takeaway: The slope of x^3 at x = 1 is 3, meaning the curve rises 3 units for every 1 unit to the right.

Understanding Curve Slopes

Tips for working with curve slopes:

Remember: slope of a curve = derivative evaluated at a point
Positive slope means the curve is going up; negative means going down
Where the slope is zero, the curve has a horizontal tangent (potential max/min)

Frequently Asked Questions about Slope of a Curve Calculator

How is the slope of a curve different from the slope of a line?

A line has a constant slope (the same everywhere). A curve's slope varies point to point. The derivative f'(x) gives the slope at each point x. At any specific point x0, the slope of the curve equals the slope of the tangent line.

When is the slope of a curve zero?

The slope of a curve is zero at critical points where f'(x) = 0. These points correspond to local maxima, local minima, or inflection points of the function.