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Chapter 1: Problem 34

Differentiate each function. $$f(x)=6 x^{-4}\left(6 x^{3}+10 x^{2}-8 x+3\right)$$

Short Answer

Expert verified
f'(x) = -36x^{-2} - 120x^{-3} + 144x^{-4} - 72x^{-5}.

Step by step solution

01

- Differentiate the Product of Functions

Use the product rule, \( (u \cdot v)' = u' \cdot v + u \cdot v' \,\){where}\ u = 6x^{-4} and v = 6x^{3} + 10x^{2} - 8x + 3. Differentiate these functions separately.
02

- Differentiate u

Find the derivative of \[ u = 6x^{-4}, \ so \ u' = \frac{d}{dx}(6x^{-4}) = 6 \cdot (-4)x^{-5} = -24x^{-5}. \]
03

- Differentiate v

Find the derivative of \[v = 6x^3 + 10x^2 - 8x + 3. \ Using the power rule, v' = 18x^2 + 20x - 8. \]
04

- Apply the Product Rule

Combine the differentiated functions using the product rule: \[ (u \cdot v)' = u' \cdot v + u \cdot v'. \] \[ Substitute the values from steps 2 and 3: \] \[ f'(x) = (-24x^{-5})(6x^3 + 10x^2 - 8x + 3) + (6x^{-4})(18x^2 + 20x - 8). \]
05

- Simplify the Expression

Simplify the expression: \[ f'(x) = (-144x^{-2} - 240x^{-3} + 192x^{-4} - 72x^{-5}) + (108x^{-2} + 120x^{-3} - 48x^{-4}). \] \[ Combine like terms: \] \[ f'(x) = (-144 + 108)x^{-2} + (-240 + 120)x^{-3} + (192 - 48)x^{-4} - 72x^{-5}. \] \[ Resulting in: f'(x) = -36x^{-2} - 120x^{-3} + 144x^{-4} - 72x^{-5}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule in Differentiation
When working with functions that are the product of two other functions, the product rule helps us find the derivative. The product rule states that if you have two functions, say \(u(x)\) and \(v(x)\), their product鈥檚 derivative is given by: \((u \times v)' = u' \times v + u \times v'\). In simpler terms, you differentiate the first function, keep the second function the same, then add it to the first function kept the same and differentiate the second one. In our given exercise, \(f(x) = 6x^{-4}(6x^3 + 10x^2 - 8x + 3)\), to find \(f'(x)\), we'll use the product rule.
Power Rule in Differentiation
The power rule is one of the simplest rules in differentiation, making it essential for anyone learning calculus. If you have a term like \(x^n\), the power rule states that the derivative of this term is \(nx^{n-1}\). For example, the derivative of \(6x^{-4}\) is given by multiplying 6 by the exponent -4, and then decreasing the exponent by 1. This results in: \(6 \times (-4)x^{-4-1} = -24x^{-5}\). This rule simplifies many differentiation problems. In our exercise, the power rule helps differentiate both parts of the function \(u(x) = 6x^{-4}\) and \(v(x) = 6x^3 + 10x^2 - 8x + 3\).
Introduction to Calculus
Calculus is a branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It's divided into differential and integral calculus. Differential calculus deals with the concept of a derivative, which represents the rate of change of a function. This is crucial for understanding motion, growth, and decay in physics and biology. In our exercise, we focus on differentiation using rules like the product and power rule. This allows us to break down complex functions into simpler parts that we can work with to understand their behavior fully.

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Most popular questions from this chapter

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