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Chapter 1: Problem 22

$$ \text { Differentiate. } $$ $$ y=\frac{2}{x+1} $$

Short Answer

Expert verified
The derivative is \( y' = \frac{-2}{(x+1)^2} \).

Step by step solution

01

Rewrite the Function

Rewrite the given function in a form that is easier to differentiate. From the problem, we know that: ... To make it simpler to differentiate, convert the function into a power of x:\[ y = \frac{2}{x+1} \rightarrow y = 2(x+1)^{-1} \]
02

Apply the Power Rule

Differentiate the function using the power rule which states that \( \frac{d}{dx}[x^n] = n x^{n-1} \). Here.\(y = 2(x+1)^{-1}\) can be written as applied as:\[ \frac{d}{dx}[ y = 2(x+1)^{-1}] \]. Using the power rule: \[dy/dx = 2 \times -1 (x+1)^{-2} \]. Thus, after simplification we get: \[ dy/dx = -2 (x+1)^{-2} \].
03

Simplify the Expression

Simplify the derivative by rewriting the expression back to its original form if required: \[y' = \frac{-2}{(x+1)^2} \]

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Key Concepts

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

Power Rule
The power rule is one of the most commonly used rules in differentiation. It states that if you have a function in the form of \(x^n\), the derivative of this function is \(nx^{n-1}\). This rule makes differentiation simpler and quicker.
When applying the power rule, always remember these points:
  • Identify the exponent \(n\).
  • Multiply the function by \(n\).
  • Subtract 1 from the exponent \(n\).

For example, if you have \(y = x^3\), its derivative is \(3x^{2}\). This rule helps us quickly find the slope of functions that can be expressed as powers of \x\.
Simplification
Simplification is an essential part of solving mathematical problems. In differentiation, it often means rewriting functions in a form that is easier to work with.
In our example, the function \(y=\frac{2}{x+1}\) was simplified to \(y = 2(x+1)^{-1}\) to make use of the power rule.
Remember these simplification tips:
  • Convert fractions to negative exponents to make differentiation easier.
  • Combine constants into simpler forms.
  • Use algebraic techniques like factoring or expanding to simplify expressions.

By simplifying functions first, you can apply differentiation rules more effectively and find solutions faster.
Derivative
A derivative represents the rate at which a function changes. In other words, it tells us the slope of the function at any given point.
For our example, the original function was \(y=\frac{2}{x+1}\). The steps were:
  • First, rewrite \(y = 2(x+1)^{-1}\).
  • Next, apply the power rule: \(dy/dx = -2(x+1)^{-2}\).
  • Finally, simplify back to fractional form: \(y' = \frac{-2}{(x+1)^2}\).

By differentiating, we can analyze how the function behaves at any value of \x\. This is critical in many fields such as physics, engineering, and economics for understanding trends and making predictions.

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Most popular questions from this chapter

The functions in Exercises 21-26 are defined for all \(x\) except for one value of \(x\). If possible, define \(f(x)\) at the exceptional point in a way that makes \(f(x)\) continuous for all \(x\). \(f(x)=\frac{\sqrt{9+x}-\sqrt{9}}{x}, x \neq 0\)

Each limit in Exercises 49-54 is a definition of \(f^{\prime}(a)\). Determine the function \(f(x)\) and the value of \(a\). \(\lim _{h \rightarrow 0} \frac{(1+h)^{-1 / 2}-1}{h}\)

Let \(f(p)\) be the number of cars sold when the price is \(p\) dollars per car. Interpret the statements \(f(10,000)=200,000\) and \(f^{\prime}(10,000)=-3\).

Compute the following limits. \(\lim _{x \rightarrow-\infty} \frac{1}{x^{2}}\)

Use limits to compute the following derivatives. \(f^{\prime}(0)\), where \(f(x)=x^{2}+2 x+2\)
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