What is a derivative?

Hook

You're standing on a curve and can only feel the ground right under your feet — no view of the whole shape. Which way is downhill?

Intuition

Drag the point along the curve. The dashed line is its tangent — tilt down to the right means the curve is falling there; tilt up means it's rising.

Formalize

That tilt has a name: the derivative. It's the instantaneous rate of change of $f$ at the point $x$ —

\frac{df}{dx}

$df/dx$ — the derivative of $f$ at the point $x$ : how fast $f$ changes as $x$ nudges slightly.
$f(x)$ — the function whose steepness we're measuring.
$x$ — the point along the curve where the derivative is evaluated.

Differentiate this specific curve
For the curve above, $f(x) = x^2 - 4x + 5$ . Differentiating term by term gives:
$\frac{df}{dx} = 2x - 4$
Read the sign
Its sign tells you the direction:
- positive means the curve rises as $x$ grows
- negative means it falls
- zero means you've hit a flat spot

Play

f'(5.00) = 2(5.00) − 4 = 6.00

Drag again — now watch the formula's value track the tangent line exactly. The derivative is the tangent's slope, nothing more.

Worked example

Evaluate the derivative at x = 5
$f'(5) = 2(5) - 4 = 6$
Read the sign
Positive, so the curve is rising there — to decrease $f$ , move in the negative $x$ direction. This one fact — "the derivative points toward increase, so step against it" — is the entire idea behind gradient descent, several chapters from now.

Checkpoint

Drag the point until the derivative reads (approximately) zero — that’s the flat spot where the curve stops rising or falling.

f'(5.00) = 6.00

Drag the point to try it

Summary

\frac{df}{dx}

The derivative is the slope of the tangent line at a point: it points in the direction the function increases fastest, and it's exactly zero at a flat spot like a minimum. The next chapter generalizes this same idea to functions of several variables at once — the gradient.