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Slope of x² at x = 1
- 1Pick a second point a step h away Q = (1+h, (1+h)²)
- 2Rise over run between the points ((1+h)² − 1) / h = 2 + h
- 3Shrink the gap: h → 0 2 + h → 2
- 4The limit is the tangent slope f′(1) = 2
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To find the slope of the curve at x = 1 we consider the limit of the difference quotient as h approaches zero. Expanding (1+h)² gives 1 + 2h + h², subtracting f(1) = 1 leaves 2h + h², and dividing by h yields 2 + h, which tends to 2, therefore the derivative of x² evaluated at one equals two, as can also be verified using the power rule d/dx xⁿ = n·xⁿ⁻¹, a result which generalises… To find the slope of the curve at x = 1 we consider the limit of the difference quotient as h approaches zero. Expanding (1+h)² gives 1 + 2h + h², subtracting f(1) = 1 leaves 2h + h², and dividing by h yields 2 + h, which tends to 2, therefore the derivative of x² evaluated at one equals two, as can also be verified using the power rule d/dx xⁿ = n·xⁿ⁻¹, a result which generalises…
Slide the second point into the first, and the slope you measure becomes the slope at that point.
- 1Pick a nearby point
- 2Rise over run = 2 + h
- 3Let h shrink to 0
- 4Slope settles: f′(1) = 2
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Anything you can screenshot: a homework problem, a textbook page, a lecture slide, a graph or a definition. It shines on math and science (calculus, algebra, physics, statistics, chemistry) and also handles economics and computer science.
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