Gradient/Slope Calculator – Find Gradient at a Point

Calculate the gradient or slope of a line or curve with our free online calculator. Find gradients from two points or as the derivative of a function at any point.

Point 1: (x₁, y₁)

Point 2: (x₂, y₂)

Examples:

Understanding Gradient and Slope

Gradient (or slope) measures how steep a line or curve is. For a straight line, it's the ratio of vertical change (rise) to horizontal change (run). For a curve, the gradient at a point is the slope of the tangent line – the line that just touches the curve at that point.

Positive gradient means the line goes uphill (left to right). Negative gradient means downhill. Zero gradient is horizontal. Undefined gradient (division by zero) is a vertical line.

How to Calculate Gradient

From Two Points

m = (y₂ - y₁) / (x₂ - x₁)

Find the difference in y-values (rise)
Find the difference in x-values (run)
Divide rise by run

From a Function

m = f'(x) = dy/dx

Find the derivative of the function
Evaluate at the given x-value
This gives the instantaneous rate of change

Worked Examples

Example 1: Gradient from Two Points

Points: (1, 2) and (4, 8)

Rise = 8 - 2 = 6

Run = 4 - 1 = 3

Gradient = 6 / 3 = 2

For every 1 unit right, go up 2 units

Example 2: Negative Gradient

Points: (-3, 5) and (2, -1)

Rise = -1 - 5 = -6

Run = 2 - (-3) = 5

Gradient = -6 / 5 = -1.2

Negative slope: line goes downhill

Example 3: Gradient of y = x² at x = 2

f(x) = x²

f'(x) = 2x (derivative)

f'(2) = 2(2) = 4

Point: (2, 4) since f(2) = 4

Tangent: y - 4 = 4(x - 2)

Tangent equation: y = 4x - 4

Example 4: Gradient of sin(x) at x = 0

f(x) = sin(x)

f'(x) = cos(x)

f'(0) = cos(0) = 1

At x=0, sine curve has gradient 1 (45° angle)

Quick Fact

The concept of gradient as "rise over run" dates back to ancient builders and surveyors. However, the mathematical formalization of finding gradients of curves (calculus) was independently developed by Newton and Leibniz in the 1670s, revolutionizing mathematics and science.

Frequently Asked Questions

What does a gradient of 0 mean?

Zero gradient means a horizontal line – no rise, only run. The y-value stays constant. For curves, zero gradient indicates a maximum, minimum, or inflection point.

What is an undefined gradient?

Undefined gradient occurs with vertical lines where x₁ = x₂, causing division by zero. The line goes straight up and down – infinite steepness.

How does gradient relate to angle?

Angle = arctan(gradient). A gradient of 1 gives 45°. Gradient of 0 gives 0°. As gradient approaches infinity, angle approaches 90°.

What's the difference between gradient and slope?

They're the same thing! "Slope" is more common in American English, "gradient" in British English. In multivariable calculus, gradient has a more specific meaning (a vector of partial derivatives).

How is gradient used in real life?

Road signs show gradient as percentages (10% grade = 0.1 gradient). Roof pitch, wheelchair ramp requirements, and ski slope difficulty all use gradient. In economics, gradient represents marginal rates.

Can gradient be greater than 1?

Absolutely! A gradient of 2 means rising 2 units for every 1 unit across. Steep hills, roller coasters, and rocket trajectories can have gradients much greater than 1.

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