Derivative of x²
The derivative of x² is the "hello world" of calculus — the first place most students ever apply the power rule.
Step-by-step solution
You are differentiating f(x) = x2: variable base, constant exponent n = 2. That's the power rule's home turf.
d/dx[xn] = n·xn−1: bring the exponent down as a coefficient, then reduce the exponent by one.
f′(x) = 2x1 = 2x.
Why it works
You can verify this one from scratch with nothing but algebra: expand (x + h)² = x² + 2xh + h², subtract x², divide by h to get 2x + h, and let h shrink to zero. The surviving term is 2x. Doing this once by hand — even just once in your life — makes the power rule feel like a fact rather than a spell.
Geometrically, 2x says the parabola gets steeper in direct proportion to how far you are from the origin: slope −4 at x = −2, slope 0 at the vertex, slope +4 at x = +2. The perfect symmetry of the parabola shows up as the perfect oddness of its derivative. In physics, if x² is a position (constant acceleration), 2x is the velocity growing linearly with time.
Common mistakes
- Leaving the exponent unchanged: writing 2x² instead of 2x. Both parts of the rule must fire — coefficient down and exponent reduced.
- Applying the rule to 2^x, where the exponent is the variable — that needs the exponential rule instead.
- Forgetting that the rule works on each term separately: d/dx[x² + x²] = 4x, not (2x)².
Practice problems
Differentiate 5x²
Answer: 10x.
Differentiate x² − 3x + 7
Answer: 2x − 3.
At which x does x² have slope 10?
Answer: x = 5, since 2x = 10.