Differentiation yields what geometric interpretation in a function's graph?

Differentiation yields what geometric interpretation in a function's graph?

• A. The area under the curve.
• B. The slope of the tangent line.
• C. The average value of f.
• D. The maximum value of f.

Answer: (B)

Differentiation gives the slope of the tangent line to a function’s graph at a specific point. The derivative at x0, f'(x0), tells you how fast the function value is changing as x changes right near x0, which is exactly the slope of the line that just touches the curve there. That tangent line has slope f'(x0) and passes through the point (x0, f(x0)); its equation is y = f(x0) + f'(x0)(x - x0). Area under the curve comes from integration, not differentiation. The average value of f involves an integral divided by an interval length, and the maximum value relates to extrema, which are analyzed via derivatives but aren’t the direct geometric interpretation of differentiation itself.

Differentiation gives the slope of the tangent line to a function’s graph at a specific point. The derivative at x0, f'(x0), tells you how fast the function value is changing as x changes right near x0, which is exactly the slope of the line that just touches the curve there. That tangent line has slope f'(x0) and passes through the point (x0, f(x0)); its equation is y = f(x0) + f'(x0)(x - x0).

Area under the curve comes from integration, not differentiation. The average value of f involves an integral divided by an interval length, and the maximum value relates to extrema, which are analyzed via derivatives but aren’t the direct geometric interpretation of differentiation itself.