What is differentiation?

What is differentiation?

• A. The area under a curve.
• B. The value of f at a point.
• C. The sum of a series.
• D. The rate at which a function changes; defined as the limit of [f(x+h)-f(x)]/h as h->0.

Answer: (D)

Differentiation measures how fast a function changes at a point, giving the slope of the graph there—the instantaneous rate of change. The derivative is defined as the limit of the average rate of change as the input shift h goes to zero: lim_{h→0} [f(x+h) − f(x)]/h. This limit, when it exists, captures the exact slope of the tangent line to the curve at x, meaning it tells you how small changes in x produce changes in f(x) in the tiniest possible way. Think of velocity as the derivative of position with respect to time: it’s the instantaneous rate at which position changes. By contrast, the area under a curve is what you compute with integration, not differentiation, and the value of f at a point is simply the function's output at that input. For a quick intuition, if f(x)=x^2, the derivative is 2x, which tells you the steepness of the curve at any x.

Differentiation measures how fast a function changes at a point, giving the slope of the graph there—the instantaneous rate of change. The derivative is defined as the limit of the average rate of change as the input shift h goes to zero: lim_{h→0} [f(x+h) − f(x)]/h. This limit, when it exists, captures the exact slope of the tangent line to the curve at x, meaning it tells you how small changes in x produce changes in f(x) in the tiniest possible way.

Think of velocity as the derivative of position with respect to time: it’s the instantaneous rate at which position changes. By contrast, the area under a curve is what you compute with integration, not differentiation, and the value of f at a point is simply the function's output at that input. For a quick intuition, if f(x)=x^2, the derivative is 2x, which tells you the steepness of the curve at any x.