Write a package that defines a function diff[expr, var ] to differentiate expressions symbolical

step 1 Hello, so here we have our function, G of x is equal to x plus 1 times e to the x over x minus 2

Write a package that defines a function diff[expr, var ] to...

Write a package that defines a function $\operatorname{diff}[\operatorname{expr}$, var $]$ to differentiate expressions symbolically. The derivative of an arithmetic expression $U$ with respect to $x$, $\left(\frac{d U}{d x}\right)$, is defined recursively by differentiation rules that are applied to $U$. From a handbook, we take the following rules ( $U$ and $V$ are arbitrary arithmetic expressions, and $c$ is a constant): $$ \begin{array}{ll} \frac{d c}{d x} & \rightarrow 0 \\ \frac{d x}{d x} & \rightarrow 1 \\ \frac{d-U}{d x} & \rightarrow-\frac{d U}{d x} \\ \frac{d U+V}{d x} & \rightarrow \frac{d U}{d x}+\frac{d V}{d x} \\ \frac{d U V}{d x} & \rightarrow \frac{d U}{d x} V+U \frac{d V}{d x} \\ \frac{d U^c}{d x} & \rightarrow c U^{c-1} \frac{d U}{d x} \\ \frac{d \exp (U)}{d x} & \rightarrow \exp (U) \frac{d U}{d x} \\ \frac{d \ln (U)}{d x} & \rightarrow \frac{1}{U} \frac{d U}{d x} \\ \frac{d U^V}{d x} & \rightarrow U^V \frac{d V \ln (U)}{d x} \end{array} $$ 1. Write these rules as definitions for $\operatorname{diff}[]$. Not all of these rules are strictly necessary. Which ones are sufficient to differentiate all the functions given? 2. These definitions can be augmented by rules for special functions, such as sine, cosine, and so on. The function diff [] becomes more powerful in this way. Implement a few of these rules.

Write a package that defines a function $\operatorname{diff}[\operatorname{expr}$, var $]$ to differentiate expressions symbolically.
The derivative of an arithmetic expression $U$ with respect to $x$, $\left(\frac{d U}{d x}\right)$, is defined recursively by differentiation rules that are applied to $U$. From a handbook, we take the following rules ( $U$ and $V$ are arbitrary arithmetic expressions, and $c$ is a constant):
$$
\begin{array}{ll}
\frac{d c}{d x} & \rightarrow 0 \\
\frac{d x}{d x} & \rightarrow 1 \\
\frac{d-U}{d x} & \rightarrow-\frac{d U}{d x} \\
\frac{d U+V}{d x} & \rightarrow \frac{d U}{d x}+\frac{d V}{d x} \\
\frac{d U V}{d x} & \rightarrow \frac{d U}{d x} V+U \frac{d V}{d x} \\
\frac{d U^c}{d x} & \rightarrow c U^{c-1} \frac{d U}{d x} \\
\frac{d \exp (U)}{d x} & \rightarrow \exp (U) \frac{d U}{d x} \\
\frac{d \ln (U)}{d x} & \rightarrow \frac{1}{U} \frac{d U}{d x} \\
\frac{d U^V}{d x} & \rightarrow U^V \frac{d V \ln (U)}{d x}
\end{array}
$$
1. Write these rules as definitions for $\operatorname{diff}[]$. Not all of these rules are strictly necessary. Which ones are sufficient to differentiate all the functions given?
2. These definitions can be augmented by rules for special functions, such as sine, cosine, and so on. The function diff [] becomes more powerful in this way. Implement a few of these rules.

Key Concepts

Special Function Differentiation

Special function differentiation involves extending the basic rules to encompass functions beyond arithmetic operations, such as exponential, logarithmic, trigonometric, and other transcendental functions. By defining rules for these functions, the differentiation package can handle a wider variety of expressions, making it more versatile and powerful for practical applications.

Symbolic Differentiation

Symbolic differentiation is the process of finding the derivative of an expression exactly, in terms of symbols, rather than through numerical approximation. This process involves applying algebraic manipulation and calculus rules to compute derivatives systematically and is fundamental to computer algebra systems.

Recursive Rule Application

Recursive rule application is a method where differentiation rules are applied repeatedly to the components of an arithmetic expression until all the elements have been reduced to basic derivatives. This approach leverages the structure of the expression, breaking it down into simpler parts that can be handled individually.

Basic Differentiation Rules

Basic differentiation rules include the derivatives of constants, variables, sums, and products. These rules, such as the constant rule (derivative of a constant is zero), the variable rule (derivative of a variable with respect to itself is one), the sum rule (derivative of a sum is the sum of the derivatives), and the product rule, form the foundation for differentiating most arithmetic expressions.

Chain Rule

The chain rule is essential when differentiating composite functions, where one function is applied within another. It provides a systematic way to take the derivative of a composition by differentiating the outer function and multiplying it by the derivative of the inner function, extending the capability of symbolic differentiation to handle complex nested expressions.

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