Problem 34 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=\left(x^{5}-3 x\right)\left(\frac{1}{x^{2}}\right) $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=(x^{5}-3x) * \frac{1}{x^{2}}\) is \(f'(x)= 3x^{2}+6x^{-1}-3x^{-2}\). The product rule was used to find this derivative.

Step 1: Identify the function types

The given function can be expressed as a product of two simpler functions h(x) and g(x). Let's identify each function: \(h(x) = x^{5}-3x\) and \(g(x) = \frac{1}{x^{2}}\) which can also be written \(g(x) = x^{-2}\) for simpler calculations.

Step 2: Apply the Product Rule

The product rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function. Therefore, applying the product rule on \(f(x) = h(x) * g(x)\), we find that \(f'(x)=h'(x)*g(x)+h(x)*g'(x)\). Now we need to find the derivatives of \(h(x)\) and \(g(x)\).

Step 3: Calculate the first derivatives

The derivative of \(h(x)=x^{5}-3x\) is \(h'(x)=5x^{4}-3\), applying the power rule. The derivative of \(g(x)=x^{-2}\) is \(g'(x)=-2x^{-3}\), also applying the power rule.

Step 4: Substitute back into the product rule

Substituting the derived functions into the product rule gives: \(f'(x)= (5x^{4}-3)*x^{-2} + (x^{5}-3x)*(-2x^{-3})\).

Step 5: Simplify the equation

Simplifying the equation, we get: \(f'(x)= 5x^{2}-3x^{-2}-2x^{2}+6x^{-1} = 3x^{2}+6x^{-1}-3x^{-2}\)

Key Concepts

Understanding the Product Rule

The product rule is a key differentiation technique used when dealing with the derivative of a product of two functions. It can seem daunting at first, but it's simply a matter of breaking things down. If you have two functions multiplied together, say \( h(x) \) and \( g(x) \), you can find their derivative using this rule:
\[ f'(x) = h'(x)g(x) + h(x)g'(x) \]
This means you take the derivative of the first function, multiply it by the second function, then add the product of the first function and the derivative of the second one.

• Identify each function in the product.
• Find the derivative of each function separately.
• Apply the rule by following the formula.

In our example, the function \( f(x) \) is a product of \( h(x) = x^5 - 3x \) and \( g(x) = \frac{1}{x^2} \). So, we first needed to find \( h'(x) \) and \( g'(x) \) before substituting them into the product rule formula.

Exploring the Power Rule

The power rule is fundamental in calculus when differentiating functions of the form \( x^n \). The power rule states that if \( y = x^n \), then the derivative \( y' \) is \( nx^{n-1} \). This rule makes taking derivatives of polynomials straightforward.

• Focus on the exponent of \( x \).
• Multiply by the current exponent.
• Decrease the exponent by one.

For our function, \( h(x) = x^5 - 3x \) and \( g(x) = x^{-2} \), we applied the power rule:
- \( h'(x) = 5x^{4} - 3 \) where \(-3x\) is treated as \( -3x^1 \) and its derivative is \(-3\).- \( g'(x) = -2x^{-3} \), where taking the derivative of \( x^{-2} \) involves multiplying by -2 and reducing the power by one.

The Differentiation Process

Differentiation is the overall process of finding the derivative of a function, which gives us the rate at which the function's value changes. By leveraging different rules such as the product rule and power rule, differentiation becomes systematic and logical.
Starting with function identification, you break down complex expressions into simpler components. For our problem:

• Recognize the expression as a product: \( (x^5 - 3x)(\frac{1}{x^2}) \).
• Apply the product rule using the derivatives obtained by the power rule.
• Simplify the final expression after substituting derivatives back into the equation.

After applying these steps, we simplified the terms to get the final derivative: \( f'(x)= 3x^2 + 6x^{-1} - 3x^{-2} \). Understanding these rules is key to mastering differentiation.