Problem 10 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=\frac{1}{\left(x^{5}-x+1\right)^{9}} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -9 (5x^4 - 1)(x^5 - x + 1)^{-10} \).

Step 1: Identify the Function Type

The function given is a rational function with a denominator that is a polynomial raised to a power. Specifically, it takes the form \( f(x) = \frac{1}{(g(x))^9} \) where \( g(x) = x^5 - x + 1 \).

Step 2: Recall the Chain Rule and Power Rule

To differentiate this function, we need to apply the chain rule and the power rule. The chain rule states that if you have a composite function \( f(g(x)) \), then its derivative is \( f'(g(x)) \, g'(x) \). The power rule states that the derivative of \( x^n \) is \( n x^{n-1} \).

Step 3: Differentiate Using the Chain Rule

Set \( u = g(x) = x^5 - x + 1 \). Then \( f(x) = u^{-9} \). Applying the chain rule: \[ f'(x) = \frac{d}{dx} (u^{-9}) \cdot \frac{du}{dx} \] First, find the derivative of \( u^{-9} \) with respect to \( u \), which is \( -9 u^{-10} \).

Step 4: Differentiate \( g(x) \)

Find \( g'(x) \), the derivative of \( x^5 - x + 1 \).\[ g'(x) = \frac{d}{dx} (x^5) - \frac{d}{dx} (x) + \frac{d}{dx} (1) = 5x^4 - 1 \]

Step 5: Combine Results Using Chain Rule

Substitute back into the chain rule: \[ f'(x) = -9 (x^5 - x + 1)^{-10} \, (5x^4 - 1) \] This is the expression for \( f'(x) \).

Step 6: Simplify the Derivative Expression

Thus, the derivative of the given function is: \[ f'(x) = -9 (5x^4 - 1)(x^5 - x + 1)^{-10} \] Ensure all parentheses are clearly denoted in the final expression to avoid errors.

Key Concepts

Chain Rule

When faced with a composite function, like the one in the problem, the chain rule becomes a crucial tool. You might have a function embedded within another function, such as \( f(g(x)) \). The chain rule offers a systematic method to find the derivative of these intertwined functions.

• Imagine peeling a layered onion. This rule allows you to differentiate the outer layer, then multiply by the derivative of the inner layer.
• The essence of the chain rule is in recognizing the layers and systematically handling each one.

In our exercise, we recognized \( f(x) = (g(x))^{-9} \) as a composite function where \( g(x) = x^5 - x + 1 \). By identifying \( g(x) \), and using the chain rule, we first differentiated the outer function, \( u^{-9} \), with respect to \( u \), finding it to be \( -9 u^{-10} \). Then, we multiplied this by the derivative of the inner function, \( g(x) \), giving us the complete derivative. After these steps, differentiating complex functions becomes a choreographed dance of consistency and precision.

Power Rule

The power rule is one of the foundational rules in calculus differentiation. It's quite simple and efficient, especially when dealing with polynomials. The rule states that the derivative of \( x^n \) is \( n x^{n-1} \). Let's see how it works:

• The power rule helps simplify complex expressions by reducing the power of \( x \) while multiplying by the original power exponent.
• This rule is extremely useful, as it applies not just to simple terms like \( x^2 \), but also to more complex ones seen in polynomials.

In our step-by-step solution, we used the power rule when we differentiated \( g(x) = x^5 \) which resulted in \( 5x^4 \). This was a key part of finding \( g'(x) \), which was necessary for completing the differentiation process of the composite function using the chain rule. The power rule works seamlessly in tandem with other differentiation rules to unpack even the most difficult functions.

Polynomial Derivative

Polynomials are prevalent in calculus, and understanding how to differentiate them efficiently is essential. The derivative of a polynomial is obtained by applying the power rule to each individual term. Here's what it entails:

• Every term in the polynomial gets differentiated separately, after which all resulting terms are summed.
• The derivatives of constants are zero since constants do not change and thus don't contribute to slopes.

In the given exercise, the function \( g(x) = x^5 - x + 1 \) posed as our polynomial. We derived \( g'(x) = 5x^4 - 1 \), breaking each term down:

• For \( x^5 \), we applied the power rule, bringing down the exponent 5 to get \( 5x^4 \).
• For \( -x \), a similar rule applies resulting in \( -1 \).
• The constant 1 simply vanishes in differentiation to 0.

By systematically handling each piece of the polynomial, we created a simple pathway to the solution. Polynomial derivatives are like building blocks, where comprehending each piece facilitates solving larger calculus puzzles.