Question

Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule $$ \frac{d\left(u^{n}\right)}{d x}=n u^{n-1}\left(\frac{d u}{d x}\right) $$ by adding a new clause to the deriv program and defining appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol \(* *\) to denote exponentiation.) Build in the rules that anything raised to the power 0 is 1 and anything raised to the power 1 is the thing itself.

Short answer

Add a differentiation rule handling u^n with special cases for exponents 0 and 1.

Step-by-step solution

Modify the Differentiator Structure

To extend the differentiator, you begin by adding a new clause to the `deriv` procedure to handle exponentiation. Check if the expression is an exponentiation and apply the new differentiation rule: \( \frac{d(u^n)}{dx} = n \, u^{n-1} \frac{du}{dx} \).

Implement the Function exponentiation?

Define the procedure `exponentiation?` that checks if the given expression is of the form \(u^n\). This will determine if part of the expression should be differentiated using the exponentiation rule.

Define base, exponent, and make-exponentiation

The `base` function should extract the base \(u\) from the expression \(u^n\), while the `exponent` function retrieves the exponent \(n\). The function `make-exponentiation` should return the expression \(u^n\) given a base \(u\) and an exponent \(n\).

Apply Differentiation Rule for Exponentiation

Within the added clause, use the differentiation rule for exponentiation. Calculate \(n \, u^{n-1}\) using the `base` and `exponent` functions. Then recursively derive `du/dx` using the core differentiation program. Multiply these to find the derivative of \(u^n\).

Build in Special Cases for Exponents 0 and 1

Add special cases: when the exponent is 0, return 1; when the exponent is 1, return the base itself, effectively bypassing further calculations for these simplified scenarios.

Key concepts

Differentiation Rules

Differentiation rules are the foundational guidelines used to find the derivative of mathematical expressions. These rules help us determine how functions change with respect to a variable. In calculus, derivatives represent the rate of change, and therefore, understanding the standard differentiation rules is essential.

• **Power Rule**: This is one of the most common rules. For an expression of the form \(x^n\), the derivative is \(n \cdot x^{n-1}\).
• **Product Rule**: Used when differentiating products of functions. If \(f(x)\) and \(g(x)\) are functions, then the derivative of their product is \(f'(x)g(x) + f(x)g'(x)\).
• **Chain Rule**: Applied to composite functions. If \(f(g(x))\) is a composite function, its derivative is \(f'(g(x)) \cdot g'(x)\).

Differentiation rules allow us to handle a variety of mathematical problems and are critical to proficiencies in calculus and other fields.

Exponentiation

Exponentiation involves raising a number or expression to a power, representing repeated multiplication. For example, in the expression \(u^n\), \(u\) is multiplied by itself \(n\) times. It's crucial to handle this operation properly when dealing with differentiation.

• **Basic Understanding**: For any expression \(x^n\), if \(n = 0\), the result is 1. If \(n = 1\), the result is \(x\).
• **Differentiation of Exponential Functions**: The differentiation rule specific to exponentiation states that \(\frac{d(u^n)}{d x} = n \cdot u^{n-1} \cdot \frac{d u}{d x}\). This is applied when the base of the exponent \(u\) is a function itself.

Exponentiation is a vital concept in both algebra and calculus, often requiring special attention when programming symbolic differentiation.

Programming Recursion

Programming recursion is a technique where a function calls itself to solve sub-problems. It is particularly useful in the context of symbolic differentiation, where expressions can be broken down recursively until they reach a base case. Recursion simplifies the process of differentiating complex functions by handling smaller parts of the problem first.

• **Base Case**: Every recursive function needs a base case to avoid infinite loops. This could be simple forms of expressions that can be solved without further recursion.
• **Recursive Case**: The part of the function where the method calls itself to work on smaller or simpler sub-problems. In differentiation, this might involve breaking down expressions into their components and applying differentiation rules recursively.

Recursive solutions are elegant and powerful, often simplifying code when dealing with nested or repeated patterns in data or calculations.

Mathematical Expressions

Mathematical expressions are combinations of symbols representing numbers, variables, operations, and sometimes functions. They are the basic building blocks in mathematics, used to convey calculations and equations.

• **Components**: Standard elements of a mathematical expression can include constants, variables, coefficients, operators (like +, -, *, /), and functions (like sin, cos, etc.).
• **Expression Trees**: In computer science, expression trees are used to represent expressions in a structured way. Each node is an operation or operand, allowing expressions to be evaluated or transformed systematically, which is useful for symbolic differentiation.

Mathematical expressions are central to the understanding and execution of programming tasks involving calculus and algebra, especially when translating written formulas into functional code.