How x² + 2x + 5 Becomes 2x + 2

A curve, not a snapshot

Consider the function

f(x) = x² + 2x + 5

For each input x, you get a height on a smooth curve. The value alone says where you are. The derivative says how that height is changing as x moves.

Differentiate term by term

Power rule: d/dx (xⁿ) = n · xⁿ⁻¹. Constants vanish under differentiation.

d/dx (x²) = 2x
d/dx (2x) = 2
d/dx (5)  = 0

Add them:

f'(x) = 2x + 2

So the messy-looking quadratic collapses into a simple line that tracks slope.

What 2x + 2 reveals

At a point x = a, the number f'(a) = 2a + 2 is the instantaneous rate of change of f.

• If f'(a) > 0, the curve is rising through x = a.
• If f'(a) < 0, it is falling.
• If f'(a) = 0, the tangent is flat—a candidate peak or valley.

Here f'(x) = 2(x + 1) = 0 when x = -1. Near that point the quadratic bottoms out; elsewhere the sign of 2x + 2 tells you whether f grows or shrinks as x increases.

Change, made precise

The derivative is a limit of average rates:

f'(x) = lim (h → 0)  [f(x + h) - f(x)] / h

For our f, that limit evaluates to 2x + 2. Average rise-over-run becomes an exact slope—change without needing a finite step size.

Takeaway

Differentiating x² + 2x + 5 to 2x + 2 is not a magic rewrite. It extracts the curve’s local rate of change and hands it back as a new function you can read, graph, and reason with.