The power rule and basic rules

The derivative of x¹⁰ is 10x⁹. Bring the exponent down in front and lower it by one; no limit needed.

A pattern in the limits

Computing every derivative from the limit definition would be slow. Fortunately the answers follow patterns. You have already seen d/dx x = 1, d/dx x² = 2x, and the start of d/dx x³ = 3x². The power rule says that for any positive integer n, d/dx xⁿ = n xⁿ⁻¹. OpenStax proves it by expanding (x + h)ⁿ with the binomial theorem: the first term xⁿ cancels, the next is n xⁿ⁻¹ h, and every other term still contains h after dividing by h, so it vanishes in the limit.

The rule extends beyond positive integers. For negative exponents, 1/x = x⁻¹ has derivative −1·x⁻² = −1/x², matching the limit you computed for 1/x at x = 1. For fractional exponents, √x = x^1/2 has derivative (1/2)x^−1/2 = 1/(2√x). So at x = 4 the slope of √x is 1/4. Rewriting roots and reciprocals as powers before differentiating is the habit that makes this rule so useful.

Constants, multiples, sums

Three more rules follow straight from the limit laws. The derivative of a constant is 0, because a horizontal line has slope 0. A constant factor passes through: d/dx (c·f) = c·f′. A sum or difference is differentiated term by term: d/dx (f ± g) = f′ ± g′. Together with the power rule, these handle every polynomial. For p(x) = 5x³ − 2x + 7, the derivative is 15x² − 2. The constant 7 disappears because shifting a graph up or down does not change any slope.

With the rules, a tangent line takes a minute. For y = x³ at x = 2: the point is (2, 8), the slope is 3(2)² = 12, and the tangent is y = 8 + 12(x − 2) = 12x − 16. For y = x² − 6x + 1, the slope 2x − 6 is zero at x = 3, so the parabola has a horizontal tangent there, at its lowest point (3, −8). Finding where a derivative is zero is the main idea of unit 6.

Checking an answer

The rules are fast, so it pays to check them now and then against the definition or against numbers. Take f(x) = x³ at x = 2, where the power rule promises f′(2) = 12. With h = 0.01 the difference quotient is (2.01³ − 8)/0.01 = 12.0601, and with h = 0.001 it is about 12.006. The numbers close in on 12 from above, as they should. A slip such as writing 3x³ instead of 3x² would give 24, and a single difference quotient catches it at once. The lab below does the same comparison for every point you drag to.

Polynomials are also a good place to see what the derivative says about shape. The derivative of a degree n polynomial has degree n − 1, so a cubic has a quadratic slope function with at most two zeros, and a cubic can turn around at most twice. A quadratic has a linear slope function with one zero, which is why every parabola has exactly one turning point.