A derivative answers the most useful question in mathematics: 'At this exact point, how fast is this changing?' It's the slope of the curve at a single point — the steepness of the hill right under your feet. Without derivatives, gradient descent doesn't exist. Without gradient descent, deep learning doesn't exist.
What the sign of the derivative tells you
Positive derivative → function is rising. Negative derivative → function is falling. Zero derivative → you're at a local minimum or maximum. In ML, we hunt for the zero — that's where the loss is lowest.
Common derivatives to know
Power rule: d/dx (xⁿ) = n·xⁿ⁻¹
Constant: d/dx (c) = 0
Exponential: d/dx (eˣ) = eˣ
Log: d/dx (ln x) = 1/x
Sine: d/dx (sin x) = cos x
Cosine: d/dx (cos x) = −sin xMemorise the power rule and exponential — they appear in every ML derivation