The short versionA derivative tells you how fast something is changing at one exact moment: the slope of the curve right there. For powers of x, the power rule gets it in one step.
Main points
The derivative is the slope of the tangent line at a point.
It's defined as a limit: the slope between two points as they slide together.
Power rule: bring the exponent down, then subtract one from it.
2. From secant to tangent 3:40–9:55
3:52A secant line through two points becomes the tangent as the gap h shrinks to 0.
6:10Definition: f′(x) = limh→0 [f(x+h) − f(x)] / h.
9:05A constant never changes, so its derivative is 0.
IN SHORTDerivative = slope at an instant, found with a limit.
d/dxxn = n · xn−1POWER RULE · works for any real n
EXAMPLE, STEP BY STEP
Differentiate f(x) = 3x4 − 5x + 2
3x4 → 3 · 4x3 = 12x3
−5x → −5 (x1 becomes x0 = 1)
2 → 0 (it's a constant)
→ f′(x) = 12x3 − 5
!Bring the exponent down first, then subtract one. Not the other way round.
Practice quiz tap to check
What is the derivative of x5?5x4
What does f′(2) tell you?The slope of the curve at x = 2
Differentiate f(x) = 7.0, because a constant doesn't change
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