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.
For a model like a cooling temperature or a rising population, f(x) gives the amount and f′(x) gives the rate of change. 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:
sin x → cos x, cos x → −sin x (note the minus), eˣ → eˣ (unchanged!), and ln x → 1/x.
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.