what does it mean to differentiate a function

Differentiating a function means finding its derivative , which tells you how the function changes at a point. In simple terms, it gives the slope of the graph or the rate of change of the output with respect to the input.

Quick idea

If y=f(x)y=f(x)y=f(x), then differentiating tells you how fast yyy changes when xxx changes a little.

For example, if a function models distance over time, its derivative tells you speed.

What it means mathematically

The derivative is defined by a limit:

f′(a)=lim⁡h→0f(a+h)−f(a)hf'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}f′(a)=h→0lim​hf(a+h)−f(a)​

This measures the slope of the tangent line to the graph at x=ax=ax=a.

In plain language

• Before differentiation: you have the function itself.
• After differentiation: you have a new function showing how steep or fast the original one is changing.

• A positive derivative means the function is increasing, a negative derivative means it is decreasing, and zero often means a flat point.

Tiny example

If f(x)=x2f(x)=x^2f(x)=x2, then its derivative is f′(x)=2xf'(x)=2xf′(x)=2x, so the slope gets steeper as xxx gets larger.

If you want, I can also explain this with a graph or show how to differentiate a simple function step by step.