Rational Expressions As Sums Of Integer Powers

Why rewrite before differentiating?

A polynomial divided by a power of its variable can look like a single complicated fraction. But the numerator is a sum, and each numerator term can be divided by the denominator separately. Rewriting the expression this way turns it into a sum of powers, which can be differentiated term by term.

The resulting powers may be negative integers. For example, \(x^{-2}\) means \(1/x^2\), so it is another way to write a rational term. This notation makes the power rule useful for expressions whose original form has a monomial denominator.

From a rational expression to integer powers

A monomial denominator has the form \(c x^m\), where \(c\ne 0\) and \(m\) is a nonnegative integer. When \(m>0\), the original expression is defined only for \(x\ne 0\). When \(m=0\), the denominator is the nonzero constant \(c\), so it imposes no restriction on \(x\).

Write a polynomial numerator as \(a_0+a_1x+\cdots+a_nx^n\), allowing some coefficients \(a_k\) to be zero. For each term \(a_kx^k\), division by \(cx^m\) gives a coefficient \(a_k/c\) and a power \(x^{k-m}\).

If a term’s exponent \(k-m\) is negative, its power is defined for \(x\ne 0\). If \(k=m\), that term becomes a constant because \(x^0=1\). A zero numerator coefficient simply contributes no term.

Differentiate the resulting powers

The power rule states that the derivative of \(x^r\) is \(r x^{r-1}\) for a positive integer \(r\), and also for a negative integer \(r\) when \(x\ne 0\). For \(r=0\), the function is the constant \(1\), whose derivative is \(0\). The sum rule says that a finite sum can be differentiated by differentiating each term, and constant factors remain as factors.

Thus, once the rational expression is rewritten, differentiate every term using its own exponent. The resulting derivative is valid on the domain of the original expression. In particular, if the original denominator contains \(x^m\) with \(m>0\), keep \(x\ne 0\) as a domain restriction even if the rewritten expression contains a constant term.

Why the rewriting method works

The conversion is an identity on the domain of the original fraction. It follows from distributing division across a finite sum and applying the exponent law \(x^k/x^m=x^{k-m}\), which is valid when \(x\ne 0\). Differentiation then follows from the power rule and sum rule.

More explicitly, for \(x\ne 0\), the exponent law gives \(x^k/x^m=x^{k-m}\) for every nonnegative integer \(k\) and positive integer \(m\). Therefore each term in the numerator satisfies

Adding these equalities gives the rewritten expression. Applying the power rule and sum rule to that finite sum gives its derivative. This explains both why the conversion is valid and why it leads directly to a term-by-term differentiation method.

Worked examples

A common exponent error

When dividing powers with the same base, subtract the denominator exponent from the numerator exponent. Reversing the order can turn a negative power into a positive one and change the function.

Keep the original domain in view

Negative powers make the restriction at zero visible, but the restriction comes from the original denominator. For example, \(x^4/x^4\) rewrites as \(1\) for \(x\ne 0\), yet the original fraction is still undefined at zero. Do not use a simplified expression to add inputs that were excluded by the original denominator.

The method applies when the denominator is a nonzero constant times a single power of the variable. It does not turn a denominator such as \(x+2\) into integer powers by termwise division; that denominator is not a monomial. For the monomial case, the reliable sequence is: record the domain, divide each term, subtract exponents in the correct order, and differentiate the resulting sum.