Constant And Power Rules

Constant and Power Rules

Computing derivatives from first principles is tedious, so calculus provides essential shortcuts. The two most fundamental are the constant rule and the power rule.

The Constant Rule

The constant rule states that any constant function has zero derivative everywhere: $\frac{d}{dx}[c] = 0$

Graphically, $y = c$ is a horizontal line with slope zero at every point. This is intuitive: unchanging quantities have no rate of change.

The Power Rule

The power rule applies to functions of the form $x^n$ where $n$ is any real number: $\frac{d}{dx}[x^n] = n\,x^{n-1}$

The pattern is simple: multiply by the exponent, then decrease the exponent by one. For example:

• $\frac{d}{dx}[x^5] = 5x^4$
• $\frac{d}{dx}[x^2] = 2x$
• $\frac{d}{dx}[x] = 1$

Beyond Positive Integers

The power rule works for negative and fractional exponents too. With negative exponents, rewrite fractions as powers first: $\frac{d}{dx}\left[\frac{1}{x}\right] = \frac{d}{dx}[x^{-1}] = -x^{-2} = -\frac{1}{x^2}$

For roots, convert to fractional exponents: $\frac{d}{dx}[\sqrt{x}] = \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Combining the Rules

When differentiating a polynomial like $f(x) = 7x^3 - \frac{2}{x^4} + 9$, rewrite each term with exponents: $f(x) = 7x^3 - 2x^{-4} + 9$

Then apply the power rule term by term: $f'(x) = 21x^2 + 8x^{-5}$

Constants vanish, and each power term follows the exponent-down, exponent-minus-one pattern.

Key tip: Always rewrite radicals and fractions as powers before differentiating, then simplify back if needed.