Find the Fourth Derivative f(x)=(x-2)/(x-3)

Solution:

Step 1: Compute the first derivative

Step 1.1: Apply the Quotient Rule

The Quotient Rule is $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{(g(x))^2}$, where $f(x) = x - 2$ and $g(x) = x - 3$.

Step 1.2: Perform differentiation

Step 1.2.1: Use the Sum Rule

The derivative of $x - 2$ is the sum of the derivatives of $x$ and $-2$.

Step 1.2.2: Apply the Power Rule

The Power Rule states $\frac{d}{dx} x^n = n x^{n-1}$, where $n = 1$.

Step 1.2.3: Differentiate the constant

The derivative of a constant is $0$.

Step 1.2.4: Simplify the derivative

Combine like terms and simplify the expression.

Step 1.3: Finalize the first derivative

Simplify the expression to get the first derivative in its simplest form.

Step 2: Compute the second derivative

Step 2.1: Apply the Product Rule

The Product Rule is $\frac{d}{dx} (f(x)g(x)) = f(x) \frac{d}{dx} g(x) + g(x) \frac{d}{dx} f(x)$, where $f(x) = -1$ and $g(x) = \frac{1}{(x - 3)^2}$.

Step 2.2: Simplify using exponent rules

Rewrite the expression using the negative exponent rule.

Step 2.3: Apply the Chain Rule

The Chain Rule is $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$, where $f(x) = x^{-2}$ and $g(x) = x - 3$.

Step 2.4: Differentiate and simplify

Perform differentiation and simplify the expression to find the second derivative.

Step 3: Compute the third derivative

Step 3.1: Use the Constant Multiple Rule

The derivative of a constant times a function is the constant times the derivative of the function.

Step 3.2: Apply the Chain Rule

Use the Chain Rule with $f(x) = x^{-3}$ and $g(x) = x - 3$.

Step 3.3: Differentiate and simplify

Differentiate and simplify the expression to find the third derivative.

Step 4: Compute the fourth derivative

Step 4.1: Use the Constant Multiple Rule

Apply the Constant Multiple Rule to the third derivative.

Step 4.2: Apply the Chain Rule

Use the Chain Rule with $f(x) = x^{-4}$ and $g(x) = x - 3$.

Step 4.3: Differentiate and simplify

Differentiate and simplify the expression to find the fourth derivative.

Step 5: Final result

The fourth derivative of $f(x)$ with respect to $x$ is $\frac{24}{(x - 3)^5}$.

Knowledge Notes:

1. Quotient Rule: Used to differentiate functions that are ratios of two differentiable functions.

2. Sum Rule: The derivative of a sum of functions is the sum of the derivatives.

3. Power Rule: A basic differentiation rule that gives the derivative of a power function.

4. Product Rule: Used when differentiating a product of two functions.

5. Chain Rule: Used to differentiate the composition of functions.

6. Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function.

7. Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^n}$.

8. Simplification: The process of reducing a mathematical expression to its simplest form by combining like terms and applying arithmetic operations.