From a Second Derivative to Repeated Differentiation
A second derivative measures how a first derivative changes. The same idea can be repeated: differentiate the second derivative to obtain a third derivative, then continue. For powers and polynomials, this process follows a consistent pattern that makes even high-order derivatives systematic.
Repeated derivatives are useful for describing how a function changes at several levels. In this tutorial, the focus is on calculating them for powers and polynomials, including the point at which the derivatives become zero. A key detail is that the domain of the formula matters: a rule written using negative powers may not apply at \(x=0\), even when the actual derivative exists there.
The notation \(f^{(n)}\) means the \(n\)th derivative, not the \(n\)th power of \(f\). For example, \(f^{(3)}(x)=f'''(x)\). When the function is written as \(y=f(x)\), the second derivative may also be written \(\frac{d^2y}{dx^2}\).
The Repeated Power Rule
For a real exponent \(r\), the power rule gives \(\frac{d}{dx}x^r=rx^{r-1}\) for \(x>0\). Applying that rule repeatedly produces a pattern: each differentiation lowers the exponent by one and multiplies by the exponent that was present before the step.
The coefficient is called a falling product: it starts with \(r\), then uses \(r-1\), and continues for \(n\) factors. The exponent also decreases by \(n\). For instance, the third derivative has three factors, \(r(r-1)(r-2)\), and exponent \(r-3\).
Why the Falling-Product Formula Works
The pattern follows by induction on the number of differentiations. The first derivative is the power rule. Suppose the formula holds after \(n\) differentiations for \(x>0\). Differentiating that expression once more multiplies by its current exponent \(r-n\) and lowers the exponent by one.
The second line is exactly the same pattern with one additional factor and one additional decrease in the exponent. This proves the formula for every positive integer \(n\), as long as \(x>0\), where the power rule applies at each step.
Worked Example: Repeated Derivatives of a Fractional Power
Find the first four derivatives of \(g(x)=x^{5/2}\), for \(x>0\).
Differentiate once using the power rule:
Differentiate again. The new coefficient is \(\frac{5}{2}\cdot\frac{3}{2}=\frac{15}{4}\), and the exponent falls from \(\frac{3}{2}\) to \(\frac{1}{2}\):
For the third derivative, multiply by \(\frac{1}{2}\) and lower the exponent by one:
For the fourth derivative, multiply by \(-\frac{1}{2}\) and lower the exponent once more:
Check the sequence by differentiating each displayed result: the derivative of \(\frac{5}{2}x^{3/2}\) is \(\frac{15}{4}x^{1/2}\); the derivative of \(\frac{15}{4}x^{1/2}\) is \(\frac{15}{8}x^{-1/2}\); and the derivative of \(\frac{15}{8}x^{-1/2}\) is \(-\frac{15}{16}x^{-3/2}\). Each agrees with the next derivative. The domain \(x>0\) is appropriate for all four formulas.
Nonnegative Integer Powers and the Zero Derivative
A polynomial power \(x^m\), where \(m\) is a nonnegative integer, needs special attention at \(x=0\). The falling product may eventually include a zero factor, while the accompanying power may have a negative exponent. Writing that product as \(0\cdot x^{-1}\), for example, does not define a value at zero. The derivative itself is still well-defined; it is the formula’s form that must be handled correctly.
- If \(1\le n\le m\), then for all real \(x\),
$$ \frac{d^n}{dx^n}x^m = \frac{m!}{(m-n)!}x^{m-n}. $$
- If \(n>m\), then \(\frac{d^n}{dx^n}x^m=0\) for all real \(x\).
Here \(m!\), read “\(m\) factorial,” is the product of the positive integers from \(1\) through \(m\), with \(0!=1\). When \(n\le m\), the coefficient \(m!/(m-n)!\) is the same falling product \(m(m-1)\cdots(m-n+1)\). The remaining exponent is a nonnegative integer, so the expression is defined for every real \(x\).
Once \(n>m\), differentiating \(x^m\) \(m\) times produces a constant. The next derivative of that constant is zero, and all further derivatives remain zero. This explains both the formula and why the zero case applies on the entire real line.
Worked Example: Finding When a Power’s Derivatives Vanish
Let \(h(x)=x^6\). Find its derivatives through order seven and state where they are valid.
Successive differentiation gives:
Each formula through \(h^{(6)}\) has a nonnegative integer power of \(x\), so it is valid for all real \(x\). The seventh derivative is zero for all real \(x\), because the sixth derivative is the constant \(720\).
Check the last steps directly: \(\frac{d}{dx}(720x)=720\), and \(\frac{d}{dx}(720)=0\). These agree with \(h^{(6)}\) and \(h^{(7)}\), respectively.
Repeated Derivatives of Polynomials
A polynomial is a finite sum of constant multiples of nonnegative integer powers. The derivative of a sum is the sum of the derivatives, and constant multiples remain constant multiples when differentiating. Therefore, to find a polynomial’s higher derivatives, differentiate each term and then combine the results.
For a polynomial of degree \(d\), the \(d\)th derivative is a constant, and every derivative of order greater than \(d\) is identically zero. The leading coefficient affects the value of that constant, but not the order at which all subsequent derivatives vanish. Lower-degree terms may disappear sooner.
Worked Example: Repeated Derivatives of a Polynomial
Let \(p(x)=2x^5-3x^3+4x-8\). Find its derivatives through order six.
Differentiate each term. In the first derivative, \(2x^5\) gives \(10x^4\), \(-3x^3\) gives \(-9x^2\), \(4x\) gives \(4\), and the constant gives zero:
Differentiate again:
The third derivative is:
The fourth and fifth derivatives are:
Finally, the sixth derivative is the derivative of a constant:
All these expressions are valid for every real \(x\), since this is a polynomial. Check the sequence by differentiating each result: \(10x^4-9x^2+4\) differentiates to \(40x^3-18x\); that differentiates to \(120x^2-18\); then to \(240x\), then to \(240\), and then to \(0\). The results match in order, and the vanishing after order five agrees with the degree-five polynomial.
A Common Domain Error
It is tempting to apply the falling-product formula mechanically, without checking its domain. That can produce an expression that is undefined at a point where the actual derivative exists.
Worked Example: Correcting a Formula at Zero
Consider \(f(x)=x\), which is the integer power \(x^1\). A careless use of the falling-product formula for \(n=2\) gives:
This expression is undefined at \(x=0\), because \(x^{-1}=1/x\) is undefined there. It is therefore not a valid formula for the second derivative on the whole real line.
Differentiate the function directly instead:
The corrected integer-power rule says the same thing: here \(m=1\) and \(n=2>m\), so the second derivative is identically zero for all real \(x\). The issue was not that the derivative failed to exist; the issue was using a formula with a negative power outside its valid domain.
How to Work Reliably
When finding repeated derivatives, keep track of both the order and the domain. A clear calculation proceeds in stages:
For a real exponent, the repeated power formula is stated on \(x>0\). For a nonnegative integer exponent, use the all-real formulas, including the zero case.
At each step, multiply by the current exponent and reduce the exponent by one. For a polynomial, do this term by term.
A degree-\(d\) polynomial has zero derivatives at every order greater than \(d\). Verify the final steps by differentiating the last nonzero expression.
These habits help distinguish a genuine derivative from an expression that only represents it on part of its domain. They also make it easier to catch a lost factor, an exponent reduced by the wrong amount, or a derivative that should already be zero.
Check Your Understanding
For each question, keep track of the derivative order and state the domain where the result is valid.
- Find the second and third derivatives of \(x^8\).
- For \(q(x)=3x^4-2x^2+5\), find \(q''(x)\), \(q'''(x)\), and \(q^{(5)}(x)\).
- For \(r(x)=x^{3/2}\) on \(x>0\), find its first three derivatives.
- Why is \(0\cdot x^{-1}\) not a valid way to state a derivative at \(x=0\)?
- If a polynomial has degree \(d\), what can be said about every derivative of order greater than \(d\)?
