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Magazine Chapter 3: Problem 15
Find \(y^{\prime}\) (a) by applying the Product Rule and (b) by multiplying the factors to produce a sum of simpler terms to differentiate. $$y=\left(x^{2}+1\right)\left(x+5+\frac{1}{x}\right)$$
Short Answer
Expert verified
The derivative is \( y' = 3x^2 + 10x + 2 - \frac{1}{x^2} \).
Step by step solution
01
Identify the Two Factors of the Product
The expression given is \( y = (x^2 + 1)(x + 5 + \frac{1}{x}) \). Here, the two factors are \( u = x^2 + 1 \) and \( v = x + 5 + \frac{1}{x} \).
02
Differentiate Each Factor
First, find the derivative of \( u \):\[ u = x^2 + 1 \ u' = \frac{d}{dx}(x^2 + 1) = 2x\] Next, find the derivative of \( v \):\[ v = x + 5 + \frac{1}{x} \ v' = \frac{d}{dx} \left( x + 5 + \frac{1}{x} \right) = 1 - \frac{1}{x^2}\]
03
Apply the Product Rule
The Product Rule states: \((uv)' = u'v + uv'\). Apply this to find the derivative of \( y \):\[ y' = (u'v + uv') = (2x)(x + 5 + \frac{1}{x}) + (x^2 + 1)(1 - \frac{1}{x^2})\]
04
Expand and Simplify the Expression
Expand and simplify:The first part:\[2x(x + 5 + \frac{1}{x}) = 2x^2 + 10x + 2\]The second part:\[(x^2+1)(1 - \frac{1}{x^2}) = x^2 + 1 - 1 - \frac{1}{x^2} \ = x^2 - \frac{1}{x^2}\]Combine both results:\[y' = 2x^2 + 10x + 2 + x^2 - \frac{1}{x^2} = 3x^2 + 10x + 2 - \frac{1}{x^2}\]
05
Multiply the Factors Before Differentiating
Expand the original expression first: \((x^2 + 1)(x + 5 + \frac{1}{x})\). Multiply each term:- \(x^2(x) = x^3\)- \(x^2(5) = 5x^2\)- \(x^2(\frac{1}{x}) = x\)- \(1(x) = x\)- \(1(5) = 5\)- \(1(\frac{1}{x}) = \frac{1}{x}\)Combine these:\[y = x^3 + 5x^2 + x + x + 5 + \frac{1}{x} = x^3 + 5x^2 + 2x + 5 + \frac{1}{x}\]
06
Differentiate Each Term After Expansion
Differentiate each term in the expanded form:\[y' = \frac{d}{dx}(x^3) + \frac{d}{dx}(5x^2) + \frac{d}{dx}(2x) + \frac{d}{dx}(5) + \frac{d}{dx}(\frac{1}{x})\]The derivatives:- \(\frac{d}{dx}(x^3) = 3x^2\)- \(\frac{d}{dx}(5x^2) = 10x\)- \(\frac{d}{dx}(2x) = 2\)- \(\frac{d}{dx}(5) = 0\)- \(\frac{d}{dx}(\frac{1}{x}) = -\frac{1}{x^2}\)Combine these:\[y' = 3x^2 + 10x + 2 - \frac{1}{x^2}\]
07
Compare Results from Both Methods
In both methods鈥攁pplying the Product Rule and expanding before differentiating鈥攖he derivative obtained is the same: \[ y' = 3x^2 + 10x + 2 - \frac{1}{x^2} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differentiation
Differentiation is the process by which we calculate the derivative of a function. This is a fundamental operation in calculus. The basic idea is to find the rate at which a function changes at any given point. Think of it as finding out how steep a hill is at a particular spot.
To perform differentiation:
To perform differentiation:
- We use the rules of differentiation which are straightforward once you get familiar with them.
- One of the key rules is the Product Rule used when differentiating a product of two functions.
Derivatives
The concept of a derivative is foundational in calculus. It represents the rate of change of one variable with respect to another. In simpler terms, it tells us how a function behaves as its inputs change. Derivatives can also reflect the slope of a function at a specific point.
For example:
For example:
- The derivative of a constant is zero, because a constant function does not change.
- If we have a function like \(y = x^2\), its derivative \(y' = 2x\) shows us how the function value will alter with small changes in \(x\).
Calculus problems
Solving calculus problems typically involves several steps. You need a strategy to work through these problems effectively. Understanding the principles and rules is crucial.
Key strategies include:
Key strategies include:
- Yeah always follow a systematic approach. Breaking down a problem into manageable parts simplifies the process.
- Applying rules like the Product Rule, Quotient Rule, or Chain Rule can help tackle more complex expressions.
