The derivative is the gradient function: A derivative f′(x) is a new function that tells you the gradient (slope) of f at any x — how fast the output is changing.
Amount and rate: For a model like a cooling temperature or a rising population, f(x) gives the amount and f′(x) gives the rate of change.
Learn these cold: AI exams are full of 'find the rate' questions, so these few standard derivatives are worth memorising cold.
IB-style question — power rule term by term
A drone's height above the ground is modelled by h(t) = 2t³ − 5t + 7 (metres, t in seconds).
Find h′(t).
Step by step
- First write the general rule you will use on each term.
- Differentiate 2t³: bring the 3 down (3 × 2 = 6) and drop the power by one.
- Differentiate −5t: the power of t is 1, so it becomes −5 × t⁰ = −5.
- The constant +7 has gradient 0 (a flat line). Add the pieces.
Final answer
h′(t) = 6t² − 5 (the drone's vertical velocity in m s⁻¹).
Powers can be negative or fractional: The power rule still works for roots and reciprocals — just rewrite them as powers first.
For example √x = x1/2 differentiates to ½x−1/2, and 1/x = x−1 differentiates to −x−2 = −1/x².
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Four more you simply know: Beyond powers, AI models use waves (sin, cos), growth/decay (eˣ) and logs (ln x). Their derivatives are fixed facts.
Four derivatives to know
- sin x → cos x
- cos x → −sin x (note the minus)
- eˣ → eˣ (unchanged!)
- ln x → 1/x
Formula booklet: These are in the formula booklet, but knowing them saves time.
IB-style question — mixed standard derivatives
A signal is modelled by g(x) = 4eˣ + 3sin x − 2ln x.
Find g′(x).
Step by step
- Differentiate each term using the standard results; constants in front just stay.
- 4eˣ → 4eˣ (eˣ unchanged), 3sin x → 3cos x, −2ln x → −2/x.
Final answer
g′(x) = 4eˣ + 3cos x − 2/x.
Mind the minus on cos: The classic slip is forgetting that cos x differentiates to −sin x (with a minus), while sin x → +cos x has none. Write them out so you don't drop the sign.