Power rule
The derivative of x^n is n x^(n-1). Constants disappear, and you differentiate sums term by term.
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Analogy
Bring the power down to the front, then lower the power by one.
Example
4x^3 becomes 12x^2.-5x becomes -5.The constant 7 becomes 0.f'(x) = 12x^2 - 5.
In practice
The fastest way to differentiate any polynomial.
Practice
Flashcards
- What is d/dx of 4x^3 - 5x + 7?12x^2 - 5.
Questions people ask
Does the power rule work for fractional powers?
Yes. The rule holds for any real power n. The square root of x is x to the power 1/2, so its derivative is (1/2) x to the power -1/2, which is 1 over 2 times the square root of x. Rewrite roots and reciprocals as powers first.
Why does the derivative of a constant equal zero?
A constant does not change, so its rate of change is zero. Its graph is a flat horizontal line with slope zero. That is why the +7 in 4x cubed - 5x + 7 vanishes when you differentiate. Moving a graph up or down never changes its slope.