Differentiation Formulas

Differentiation formulas are simply shortcuts that give you a derivative without going through the limit every time. The power rule $\frac{d}{dx}x^n = nx^{n-1}$ looks abstract, but it's just two moves: $n$ (the exponent comes down in front) and $x^{n-1}$ (the exponent drops by one). The other rules are even simpler: $\frac{d}{dx}c = 0$ (a constant never changes, so its rate of change is zero), $(cf)' = cf'$ (a constant multiplier just tags along) and $(f \pm g)' = f' \pm g'$ (you can do a sum one term at a time). That's the whole toolkit for polynomials.

Work out a few derivatives from the definition and a pattern jumps out. $x^2$ turns into $2x$, and if you grind through $x^3$ the same way, you get $3x^2$. The exponent keeps showing up in front, one lower. That's no accident. For a whole number $n$, expanding $(x + h)^n$ gives $x^n + nx^{n-1}h$ plus terms with $h^2$ or more. Subtract $x^n$, divide by $h$, let $h \to 0$, and only $nx^{n-1}$ survives. That's the power rule, and it turns out to work for any real exponent, including negative and fractional ones.

Rewrite first. The power rule only sees powers, so turn roots and fractions into powers before differentiating:

Then apply the rules term by term. Bring the exponent down as a coefficient and lower it by one. Constants in front stay put, and constant terms on their own disappear.

A common mistake. Since sums split up so nicely, it's tempting to think products do too, and that $(fg)' = f'g'$. They don't. Try it on $x \cdot x = x^2$: the real derivative is $2x$, but $f'g'$ would give $1 \cdot 1 = 1$. For now, expand products before differentiating. The next lesson gives the actual product and quotient rules.

Every polynomial model, from a projectile's height to a cost function, can be differentiated with these rules alone, which is why they're the workhorse of applied calculus.