Rules of Differentiation: Power, Product, Quotient

Why we want rules at all

In the last rung you saw that a derivative is genuinely a limit of difference quotients: f'(x) = lim h->0 (f(x+h) - f(x)) / h. That definition is the truth, and it is worth doing by hand a few times so the idea sticks. But imagine grinding through that limit every single time you meet x^2, or sqrt(x), or 3x^5 - x + 7. Nobody does that. Mathematicians worked the limit out once for whole families of functions and packaged the answers into short rules. The rules are not a different kind of math — each one is exactly what the limit gives you, just remembered so you never have to rederive it.

The power rule, plus constants and sums

The workhorse is the power rule: to differentiate x raised to a number, bring the exponent down in front and lower it by one. In symbols, d/dx x^n = n x^{n-1}. So x^2 becomes 2x, x^5 becomes 5x^4, and x (which is x^1) becomes 1. It even works for negative and fractional powers: 1/x = x^{-1} has derivative -1 x^{-2} = -1/x^2, and sqrt(x) = x^{1/2} has derivative (1/2) x^{-1/2} = 1/(2 sqrt(x)).

Two more rules let you handle whole polynomials. The constant-multiple rule says a number stuck in front just rides along: d/dx (c f) = c f'(x), so the derivative of 3x^5 is 3 times 5x^4 = 15x^4. The sum rule says you can differentiate term by term: d/dx (f + g) = f'(x) + g'(x). And the derivative of a plain constant is 0 — a flat line has no slope. Put together, these turn any polynomial into its slope by inspection.

f(x)  = 3x^5 - x + 7
f'(x) = 15x^4 - 1 + 0
      = 15x^4 - 1

Products and quotients (the careful ones)

Here is the most common beginner trap. When two functions are multiplied, the derivative is NOT f'(x) times g'(x). That is simply false, and you can check it: if f = g = x, then f g = x^2 whose derivative is 2x, but f'g' = 1 times 1 = 1. The correct product rule is (f g)' = f'(x) g(x) + f(x) g'(x) — read it as 'differentiate the first while keeping the second, plus keep the first while differentiating the second'. Each factor takes a turn changing.

For a ratio f(x)/g(x), the quotient rule is (f/g)' = (f'(x) g(x) - f(x) g'(x)) / g(x)^2. Notice the minus sign and the order — the term with f' comes first — and the whole denominator gets squared. A memory hook: 'low d-high minus high d-low, over low squared'. It only makes sense where g(x) is not 0, since you cannot divide by zero.

1. Differentiate h(x) = x^2 sin(x) with the product rule. First factor f = x^2 with f' = 2x; second factor g = sin(x) with g' = cos(x).
2. Assemble f'g + fg': h'(x) = 2x sin(x) + x^2 cos(x). Done — no limit needed.
3. Now a quotient: q(x) = x / (x^2 + 1). With low d-high minus high d-low over low squared: q'(x) = (1 (x^2 + 1) - x (2x)) / (x^2 + 1)^2 = (1 - x^2) / (x^2 + 1)^2.

A few more derivatives, and going to second order

A handful of non-polynomial derivatives are worth memorizing, because they show up everywhere: d/dx e^x = e^x (the function that is its own slope), d/dx ln(x) = 1/x (for x > 0), d/dx sin(x) = cos(x), and d/dx cos(x) = -sin(x). Each of these was once proved straight from the limit definition; you get to just use them. Combined with the four rules above, you can now differentiate things like e^x cos(x) or (ln x)/x in a line or two.

Because the derivative of a function is itself a function, you can differentiate again. That gives the second derivative, a higher-order derivative written f''(x) or d^2y/dx^2. If the first derivative is the rate of change, the second is the rate of change of that rate. In motion it is acceleration (how fast your speed itself is changing); on a graph it is concavity (whether the curve cups upward like a smile or downward like a frown). For f(x) = x^3 we get f'(x) = 3x^2 and then f''(x) = 6x — positive when x > 0, so the curve bends upward there.