刃 · Derivatives
Scroll of the Edge
A derivative is the edge of a blade: it tells you how steep a curve is at each point.
Power Drop
For xⁿ: bring the power down in front and lower the power by 1. A number in front just comes along.
f(x) = 3x⁶ ⇒ f ′(x)
- → Bring down 6: 3 · 6 = 18
- → Lower the power: x⁵
Answer: 18x⁵
A lone constant disappears: the derivative of 7 is 0. The derivative of 5x is 5.
Practice: Try 4x³ and x² + 5x + 7.
Clear the Path
Simplify the function first, then differentiate. Dividing by x lowers every power by one.
f(x) = (4x³ + x)/x ⇒ f ′(x)
- → Simplify: 4x² + 1
- → Differentiate: 8x
Answer: 8x
Practice: Try (6x⁴ + 2x²)/x².
Know the Pair
Learn the classic pairs by heart: sin and cos chase each other.
f(x) = sin(x) ⇒ f ′(x)
- → The slope of sin is cos
Answer: cos(x)
(cos x)′ = −sin x · (eˣ)′ = eˣ · (ln x)′ = 1/x
Practice: What is the derivative of cos(x)?
Into the dojo
Practise with the Edge ′ technique in Kung Fu Math from blue belt.