DIFF

This function computes symbolic derivatives with respect to a chosen variable and derivative order.

For a function f(x), the first derivative is:

\frac{d}{dx}f(x)

and the n-th derivative is:

\frac{d^n}{dx^n}f(x)

The result is returned as a symbolic expression string.

Excel Usage

=DIFF(expression, variable, order)

expression (str, required): Expression to differentiate.
variable (str, optional, default: “x”): Variable name for differentiation.
order (int, optional, default: 1): Positive integer derivative order.

Returns (str): String representation of the symbolic derivative.

Example 1: First derivative of a polynomial

Inputs:

expression	variable	order
x*3 + 2x	x	1

Excel formula:

=DIFF("x**3 + 2*x", "x", 1)

Expected output:

"3*x**2 + 2"

Example 2: Second derivative of a polynomial

Inputs:

expression	variable	order
x4 + x2	x	2

Excel formula:

=DIFF("x**4 + x**2", "x", 2)

Expected output:

"2*(6*x**2 + 1)"

Example 3: Derivative of a trigonometric expression

Inputs:

expression	variable	order
sin(x)	x	1

Excel formula:

=DIFF("sin(x)", "x", 1)

Expected output:

"cos(x)"

Example 4: Derivative using custom variable

Inputs:

expression	variable	order
t**5	t	3

Excel formula:

=DIFF("t**5", "t", 3)

Expected output:

"60*t**2"

Python Code

from sympy import Symbol
from sympy import sympify
from sympy import diff as sympy_diff

def diff(expression, variable='x', order=1):
    """
    Compute a symbolic derivative of an expression.

    See: https://docs.sympy.org/latest/modules/core.html#sympy.core.function.diff

    This example function is provided as-is without any representation of accuracy.

    Args:
        expression (str): Expression to differentiate.
        variable (str, optional): Variable name for differentiation. Default is 'x'.
        order (int, optional): Positive integer derivative order. Default is 1.

    Returns:
        str: String representation of the symbolic derivative.
    """
    try:
        derivative_order = int(order)
        if derivative_order < 1:
            return "Error: order must be a positive integer"

        variable_symbol = Symbol(variable)
        normalized_expression = expression.replace("^", "**")
        expr = sympify(normalized_expression)
        result = sympy_diff(expr, variable_symbol, derivative_order)
        return str(result)
    except Exception as e:
        return f"Error: {str(e)}"

Online Calculator

expression *

Expression to differentiate.

variable

Variable name for differentiation.

order

Positive integer derivative order.