The slope of a tangent line to a curve at a specific point is found by evaluating the derivative of the curve's function at that point.
Here's a breakdown:
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The Derivative Represents Slope: The derivative of a function, often denoted as f'(x) or dy/dx, provides a formula for the slope of the tangent line at any point on the curve y = f(x).
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Tangent Line: A tangent line is a straight line that "touches" the curve at a single point, representing the instantaneous rate of change of the function at that point.
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Evaluating at a Point: To find the specific slope of the tangent line at a particular point (x₀, y₀) on the curve, you substitute the x-value (x₀) into the derivative function, f'(x). The result, f'(x₀), is the slope (m) of the tangent line at that point. Therefore: m = f'(x₀).
Steps to find the slope:
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Find the Derivative: Determine the derivative, f'(x), of the function y = f(x). This involves using the rules of differentiation (power rule, product rule, quotient rule, chain rule, etc.).
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Identify the Point: Determine the x-coordinate, x₀, of the point at which you want to find the slope of the tangent line. Sometimes you're given the entire point (x₀, y₀), and sometimes you need to calculate y₀ by plugging x₀ into the original function: y₀ = f(x₀).
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Evaluate the Derivative: Substitute x₀ into the derivative function, f'(x), to calculate f'(x₀). This value is the slope of the tangent line at the point (x₀, y₀).
Example:
Let's say you have the function y = f(x) = x² and you want to find the slope of the tangent line at the point (2, 4).
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Find the Derivative: The derivative of f(x) = x² is f'(x) = 2x.
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Identify the Point: We're given the point (2, 4), so x₀ = 2.
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Evaluate the Derivative: Substitute x₀ = 2 into f'(x) = 2x: f'(2) = 2 * 2 = 4.
Therefore, the slope of the tangent line to the curve y = x² at the point (2, 4) is 4.
