What Is the Approximation Formula for a Higher-Order Derivative?

A higher-order derivative can be estimated from function values near the point of interest using finite differences. The general formula depends on which points you use and how accurate you need the result to be. A simple formula works for any derivative order, while carefully chosen stencils can reduce the approximation error.

The general forward-difference formula

Let $f^{(n)}(x)$ denote the $n$th derivative, and let $h$ be a nonzero step size. The $n$th forward difference is

$$\Delta_h^n f(x)=\sum_{k=0}^{n}(-1)^{n-k}\binom{n}{k}f(x+kh).$$

Dividing by $h^n$ gives the basic approximation:

$$f^{(n)}(x)\approx \frac{1}{h^n}\sum_{k=0}^{n}(-1)^{n-k}\binom{n}{k}f(x+kh).$$

For example, the first few cases are

$$\begin{aligned} f'(x)&\approx \frac{f(x+h)-f(x)}{h},\\ f''(x)&\approx \frac{f(x+2h)-2f(x+h)+f(x)}{h^2},\\ f'''(x)&\approx \frac{f(x+3h)-3f(x+2h)+3f(x+h)-f(x)}{h^3}. \end{aligned}$$

This comes from applying Taylor expansions to the function values and combining them so that lower-order terms cancel. If the function is sufficiently smooth, the forward formula has an error of order $h$ as $h$ approaches zero. Thus, halving $h$ typically reduces the leading error by about a factor of two, provided rounding error does not become dominant.

A more accurate central formula

The general forward formula is convenient, but it uses points on just one side of $x$ and is not always the most accurate choice. For the second derivative, a common symmetric stencil is

$$f''(x)\approx \frac{f(x+h)-2f(x)+f(x-h)}{h^2}.$$

Taylor expansion about $x$ shows that its leading error is proportional to $h^2$:

$$\frac{f(x+h)-2f(x)+f(x-h)}{h^2} =f''(x)+\frac{h^2}{12}f^{(4)}(x)+O(h^4).$$

The notation $O(h^4)$ describes terms that, under suitable smoothness assumptions, shrink at least as fast as $h^4$. The central formula is therefore second-order accurate: when $h$ is halved, its leading truncation error is about one quarter as large.

Worked example: approximate a second derivative

Take $f(x)=x^4$ and estimate $f''(1)$ using the central formula with $h=0.1$. The exact derivative is $f''(x)=12x^2$, so $f''(1)=12$.

First evaluate the function at the three stencil points:

$$f(1.1)=1.1^4=1.4641,\qquad f(1)=1,\qquad f(0.9)=0.9^4=0.6561.$$

Substitute these values:

$$\begin{aligned} f''(1)&\approx \frac{1.4641-2(1)+0.6561}{(0.1)^2}\\ &=\frac{0.1202}{0.01}=12.02. \end{aligned}$$

The estimate is $12.02$, compared with the exact value $12$. In this example the error is exactly $0.02$: since $f^{(4)}(x)=24$, the leading central-formula error is $h^2f^{(4)}(1)/12=0.01\cdot24/12=0.02$.

Choosing a step size and stencil

A smaller $h$ reduces truncation error, but it does not always improve the computed answer. When $h$ is extremely small, subtracting nearly equal function values can magnify floating-point rounding error. In practice, try a few reasonable step sizes and look for estimates that stabilize. If the function can be evaluated only at points in one direction—for example, near the boundary of its domain—a forward or backward stencil may be necessary.

For higher accuracy, use more points and choose weights that cancel additional Taylor-series terms. The weights depend on the desired derivative order, accuracy, and point locations; the basic forward formula should not be assumed to have high-order accuracy just because $n$ is large.

Common mistakes

• Using the wrong power of $h$: For an $n$th derivative, the finite-difference numerator is divided by $h^n$, not just by $h$.
• Mixing up stencil formulas: The forward second-difference formula uses $x$, $x+h$, and $x+2h$. The central second-difference formula uses $x-h$, $x$, and $x+h$.
• Ignoring the error order: The basic forward formula is generally first-order accurate; the standard central second-derivative formula is second-order accurate.
• Making $h$ too small: Roundoff and cancellation can outweigh the benefit of reducing truncation error.