Derivative Rules

Last lesson, finding the slope of a curve took secants, limits and a fair amount of effort. If every derivative demanded that whole ritual, calculus would be heavy labor. The good news: you only suffer through it once — mathematicians boiled the results down into a few short rules of differentiation, and after that most derivatives can be written down at a glance.

The slope of a constant is zero

Walk as far as you like along a horizontal line and the height never changes, so the derivative of a constant is zero:

$\frac{d}{dx} \, 7 = 0$

Do not underestimate this 0: it means any constant term in a sum simply steps out during differentiation, leaving only the parts that actually move.

The power rule: bring the exponent down

For $f(x) = x^n$ , the derivative is

$f'(x) = n \, x^{n-1}$

The recipe has two lines: bring the exponent down as a coefficient, then reduce the exponent by one. Try $f(x) = x^3$ :

$f'(x) = 3x^2$

The rule predicts a slope of $3 \times 1^2 = 3$ at $x = 1$ . Talk is cheap — check it with a secant, like last time.

InteractiveSlope Checker

Point x₀0.5Step h1.5

Secant slope

2.500

Tangent slope

1.000

Drag h toward 0 and the secant hugs the tangent — that's the derivative.

Try it: the secant agrees

For $f(x) = x^3$ , start at $x = 1$ with spacing $h$ . With $h = 1$ the secant slope is $(2^3 - 1) \div 1 = 7$ ; with $h = 0.1$ it is $(1.331 - 1) \div 0.1 = 3.31$ ; with $h = 0.01$ it is $3.0301$ ... pressing down toward 3. The rule gives exactly this limiting value.

The exponent need not be a whole number: $\sqrt{x} = x^{1/2}$ has derivative $\frac{1}{2}x^{-1/2}$ — see Exponents for how fractional exponents work.

Multiply and add: derivatives play fair

Two more easy rules:

Constant multiple: stretch a curve by a factor and its slope stretches by the same factor. The derivative of $f(x) = 5x^2$ is $f'(x) = 10x$ — the 5 stays outside untouched while the power rule handles $x^2$ ;
Sum: the derivative of a sum is the sum of the derivatives. Differentiate term by term; each term minds its own business.

Putting it together: one worked example

Stack the rules to differentiate $f(x) = x^2 + 3x$ :

$x^2$ differentiates to $2x$ ;
$3x$ differentiates to $3 \times 1 = 3$ — since $x$ is $x^1$ , bringing the 1 down leaves $x^0 = 1$ .

$f'(x) = 2x + 3$

The derivative is a function in its own right: slope 3 at $x = 0$ , slope 1 at $x = -1$ ; the slope hits zero at $x = -\frac{3}{2}$ — which is exactly the bottom of this parabola. One rule, and even the location of the valley comes for free.

Each of these rules was proved once, carefully, with limits — and never needs proving again. You worked hard once in Introduction to Derivatives, and from then on every $x^n$ takes seconds. Mathematics is full of bargains like this: prove once, use forever.

Check yourself

Quick quiz

1. For f(x) = x⁵, the derivative is?
2. What is the derivative of the constant 9?
3. For f(x) = x² + 4x, the derivative is?