When approximating f'(x) numerically at a point, which concept is typically used?

When approximating f'(x) numerically at a point, which concept is typically used?

• A. Slope of the tangent line
• B. Area under the curve
• C. Slope of the secant line between x and x+h
• D. Antiderivative

Answer: (C)

Numerical differentiation relies on the difference quotient: the slope of the secant line joining the points (x, f(x)) and (x+h, f(x+h)). This secant slope, computed as [f(x+h) − f(x)]/h, estimates the instantaneous rate of change at x. As h shrinks, this slope approaches the slope of the tangent line, which is the derivative. So, to approximate f′(x) when you don’t have an exact formula, you use the secant slope between x and x+h (forward difference), and sometimes refine with central differences for better accuracy. The area under the curve relates to integration, and an antiderivative is about reversing differentiation, not about approximating a derivative at a point.

Numerical differentiation relies on the difference quotient: the slope of the secant line joining the points (x, f(x)) and (x+h, f(x+h)). This secant slope, computed as [f(x+h) − f(x)]/h, estimates the instantaneous rate of change at x. As h shrinks, this slope approaches the slope of the tangent line, which is the derivative. So, to approximate f′(x) when you don’t have an exact formula, you use the secant slope between x and x+h (forward difference), and sometimes refine with central differences for better accuracy. The area under the curve relates to integration, and an antiderivative is about reversing differentiation, not about approximating a derivative at a point.