Why Fractional Powers Need Care
Expressions involving roots often become easier to differentiate when the radicals are written as powers. For example, \(\sqrt{x}\) can be written as \(x^{1/2}\), and \(\sqrt[3]{x^2}\) can be written as \(x^{2/3}\). The power rule then gives a direct way to differentiate them.
In a product or quotient, however, rewriting the radicals is only the beginning. You must also account for the derivative of each inner function, select an appropriate sequence of product, quotient, and chain rules, and keep track of where the original expression is defined. In particular, a derivative formula may apply only on the interior of the expression's domain.
For instance, \(x^{1/2}\) requires \(x\geq 0\), whereas \(x^{1/3}\) is real for every \(x\). The denominator of a quotient must also be nonzero. These restrictions are part of the problem, not details to add after differentiating.
The Power Rule for Fractional Powers
On an interval where \(x>0\), the power rule for rational exponents is
When a fractional power has a function in its base, the chain rule supplies the derivative of that base. In particular, where \(u(x)>0\) and \(u\) is differentiable,
Here the positive-base condition is a convenient condition that makes the real power and the displayed derivative unambiguous on an open interval. If a rational power is defined on negative inputs, its derivative can also be handled on an appropriate negative interval; these examples will use positive bases when applying the rule.
Why the Fractional Power Rule Works
A short derivation explains the exponent and coefficient in the rule. First take the \(q\)-th root, with \(x>0\), and write \(g(x)=x^{1/q}\). For a small nonzero increment \(h\), factor a difference of \(q\)-th powers:
As \(h\) tends to zero, \(g(x+h)\) tends to \(g(x)=x^{1/q}\). The sum in the denominator therefore tends to \(q(x^{1/q})^{q-1}=q x^{(q-1)/q}\). It follows that
For a positive integer \(p\), differentiate \(g(x)^p\) using the integer power rule and the chain rule:
For negative integer \(p\), the same formula follows by differentiating the reciprocal of \(g(x)^{-p}\), on \(x>0\). Thus the rational power rule holds there for every integer \(p\). Applying the chain rule to the composite \(u(x)^{p/q}\) gives the stated rule for a positive inner base.
A Reliable Rule Sequence
Before calculating, rewrite the radicals and identify the outer operations. A product of two factors calls for the product rule; a quotient calls for the quotient rule; a power of an expression involving \(x\) calls for the chain rule. A factor raised to a negative power can sometimes turn a quotient into a product, but the domain restrictions must still be retained.
Require even-root radicands to be nonnegative, exclude zero from negative powers, and exclude zeros of any denominator.
Use rational exponents so the relevant power and chain rules are visible.
Apply the product or quotient rule to the outer structure, then differentiate each composite factor with the chain rule.
Check whether a domain endpoint or a zero base requires separate attention.
Worked Examples
Worked Example: A Product of Two Fractional Powers
Differentiate \(y=(x^2+1)^{3/2}(2x-1)^{1/2}\), and state where the derivative formula applies.
The first base \(x^2+1\) is always positive. The second base requires \(2x-1\geq 0\), so the function's domain is \(x\geq 1/2\). On the interior \(x>1/2\), both bases are positive, and the product and chain rules apply. Set \(A=(x^2+1)^{3/2}\) and \(B=(2x-1)^{1/2}\). Then
Therefore,
As a check, at \(x=1\), the formula gives \(3\sqrt{2}+2\sqrt{2}=5\sqrt{2}\). Directly, \(A(1)=2\sqrt{2}\), \(A'(1)=3\sqrt{2}\), \(B(1)=1\), and \(B'(1)=1\), so \(A'(1)B(1)+A(1)B'(1)=5\sqrt{2}\), as required. The formula does not apply at \(x=1/2\), where its second term is undefined; that endpoint needs separate analysis if differentiability there is in question.
Worked Example: A Quotient with a Square Root
Differentiate \(y=\frac{\sqrt{x+2}}{(x^2+1)^{3/2}}\).
The numerator requires \(x+2\geq 0\), and the denominator is always positive. Thus the function is defined for \(x\geq -2\). Use the quotient rule on the interior \(x>-2\). Let \(N=(x+2)^{1/2}\) and \(D=(x^2+1)^{3/2}\). Their derivatives are
Since \(y'=(N'D-ND')/D^2\), substitution gives
Factor \(\sqrt{x^2+1}\) from the numerator and combine the terms over \(2\sqrt{x+2}\):
At \(x=0\), this gives \(1/(2\sqrt{2})\). Directly, \(N(0)=\sqrt{2}\), \(N'(0)=1/(2\sqrt{2})\), \(D(0)=1\), and \(D'(0)=0\), so the quotient rule gives \(N'(0)/D(0)=1/(2\sqrt{2})\), confirming the result.
Worked Example: A Product and Quotient with Nested Powers
Differentiate \(y=\frac{x^{1/3}(x^2+1)^{3/2}}{(x+2)^{2/3}}\) for \(x>0\).
All three bases are positive on this interval. Write \(y=AB/C\), where \(A=x^{1/3}\), \(B=(x^2+1)^{3/2}\), and \(C=(x+2)^{2/3}\). Their derivatives are
Applying the product rule to \(AB\), followed by the quotient rule with denominator \(C\), gives
Dividing each term by the positive factors \(A\), \(B\), and \(C\) rewrites this result as
The negative sign on the \(C'/C\) term comes from differentiating a quotient. To check the calculation at \(x=1\), the original factors have values \(A=1\), \(B=2\sqrt{2}\), and \(C=3^{2/3}\). The parenthetical factor is \(1/3+3/2-2/9=29/18\). Hence the formula gives \(y'(1)=\frac{2\sqrt{2}}{3^{2/3}}\frac{29}{18}\). The rule-by-rule expression gives the same value: \(A'=1/3\), \(B'=3\sqrt{2}\), and \(C'=(2/3)3^{-1/3}\); substituting these into \(\frac{(A'B+AB')C-ABC'}{C^2}\) and simplifying gives \(\frac{2\sqrt{2}}{3^{2/3}}\frac{29}{18}\).
A Common Mistake: Losing an Inner Derivative
Rewriting a radical as a fractional power does not remove the chain rule. For example, the following calculation is wrong:
It omits the derivative of the inner function \(3x+1\), which is \(3\). The correct calculation is
The same check is needed for every nested power: after differentiating the outer power, multiply by the derivative of its base. Also retain the original domain. For example, a square root requires a nonnegative radicand even if algebraic rearrangement makes the derivative formula look simpler.
When the Method Applies
The product and quotient rules apply where their component functions are differentiable, and a quotient rule additionally requires a nonzero denominator. The fractional-power chain rule above applies on intervals where its inner base is positive. If an even-root base is zero at a domain endpoint, the displayed derivative may not exist there as a finite two-sided derivative. Do not infer endpoint differentiability merely because the original function is defined at that point.
These checks matter because they separate two questions: where the original expression is defined, and where the derivative calculation is valid. A concise final answer should make both clear. Once the domain is recorded, rewriting radicals and identifying the outer operation usually makes the rule sequence straightforward.
Check Your Understanding
For each question, keep both the rule sequence and the domain in view.
- Rewrite \(\sqrt[3]{(x^2+4)^2}\) using a rational exponent. What is its real domain?
- Differentiate \((5x-1)^{3/2}\). On what interval does the fractional-power chain rule apply with a positive base?
- For \(\frac{(x+1)^{1/2}}{x^2+4}\), state the domain and explain why the quotient rule formula is restricted to the domain's interior.
- Identify the missing factor in the incorrect derivative \(\frac{d}{dx}(2x^2+3)^{1/3}=\frac13(2x^2+3)^{-2/3}\).
- In a derivative written as \(y\left(\frac{A'}A+\frac{B'}B-\frac{C'}C\right)\) for \(y=AB/C\), explain why the last term has a minus sign.
