Derivatives · Technique guide
Derivatives in Your Head
By Robert Hansson, Kung Fu Math
A derivative measures how fast a function changes. Calculus courses spend weeks on the theory, but in practice most derivatives come from a small set of rules applied quickly and in the right order.
When you can do those rules in your head, exam questions become faster and you can spot mistakes in your written work. Each technique below shows the rule, an example and a way to check yourself.
01
The power rule
For xⁿ, bring the exponent down in front and reduce the exponent by one: the derivative of xⁿ is n·xⁿ⁻¹. This works for negative and fractional exponents too, so rewrite roots and fractions as powers first.
Worked example
f(x) = 4x³
- → Bring down 3: 3 · 4 = 12
- → Reduce the exponent: x²
Answer: f′(x) = 12x²
Practice: Differentiate 5x⁴, x⁻² and √x (write it as x^½).
02
Constants and sums
Differentiate a sum term by term. A constant on its own disappears, because it does not change. A constant factor stays in front.
Worked example
f(x) = 3x² + 7x − 9
- → 3x² → 6x
- → 7x → 7
- → −9 → 0
Answer: f′(x) = 6x + 7
Practice: Differentiate x³ − 4x² + 2x and 10 − 3x⁵.
03
Standard functions
Memorise a few pairs: eˣ stays eˣ; ln x becomes 1/x; sin x becomes cos x; cos x becomes −sin x. A quick self-check: the slope of sin x at 0 is 1, and cos 0 = 1.
Worked example
f(x) = 2 sin x + eˣ
- → 2 sin x → 2 cos x
- → eˣ → eˣ
Answer: f′(x) = 2 cos x + eˣ
Practice: Differentiate 3 ln x, cos x − eˣ and 5eˣ.
04
The chain rule
For a function inside a function, differentiate the outer function and multiply by the derivative of the inner one. Say it out loud: outside, then times inside.
Worked example
f(x) = e^(3x)
- → Outer: e^u stays e^(3x)
- → Inner: 3x → 3
- → Multiply: 3e^(3x)
Answer: f′(x) = 3e^(3x)
Practice: Differentiate (2x + 1)⁵, sin(4x) and ln(x²).
05
The product rule
For two functions multiplied together: first times the derivative of the second, plus second times the derivative of the first. (uv)′ = u·v′ + v·u′.
Worked example
f(x) = x² · eˣ
- → u = x², u′ = 2x
- → v = eˣ, v′ = eˣ
- → x²eˣ + 2xeˣ
Answer: f′(x) = eˣ(x² + 2x)
Practice: Differentiate x sin x and x³ ln x.
Common mistakes
- Forgetting the inner derivative in the chain rule. The derivative of sin(4x) is 4 cos(4x), not cos(4x).
- Treating a product like a sum. The derivative of x²·eˣ is not 2x·eˣ.
- Losing the minus sign: the derivative of cos x is −sin x.
Summary
- Power rule: bring the exponent down and reduce it by one.
- Know the derivatives of eˣ, ln x, sin x and cos x by heart.
- Chain rule: outside times inside. Product rule: u·v′ + v·u′.
Into the dojo
Reading builds understanding; practice builds speed. Train this in Kung Fu Math — no account needed.