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Chapter 1: Problem 32

$$ \text { Differentiate. } $$ $$ y=\frac{7}{\sqrt{1+x}} $$

Short Answer

Expert verified
\( y' = -\frac{7}{2}(1+x)^{-\frac{3}{2}} \)

Step by step solution

01

Rewrite the Function

Rewrite the function in a more convenient form for differentiation. Instead of the square root in the denominator, express it as a power: \[ y = 7(1+x)^{-\frac{1}{2}} \]
02

Differentiate Using the Chain Rule

Apply the chain rule to differentiate the function. The chain rule states that if you have a composite function, you differentiate the outer function and then multiply by the derivative of the inner function.Given function: \[ y = 7(1+x)^{-\frac{1}{2}} \]Step-by-step differentiation:1. Differentiate the outer function: \[ u = (1+x)^{-\frac{1}{2}} \] and its derivative with respect to \(x\) is\[ \frac{d}{dx}[(1+x)^{-\frac{1}{2}}] = -\frac{1}{2}(1+x)^{-\frac{3}{2}} \]2. Multiply by the constant and the derivative of the inside function: \[ y' = 7 \times -\frac{1}{2}(1+x)^{-\frac{3}{2}} \times 1 = -\frac{7}{2}(1+x)^{-\frac{3}{2}} \]

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Key Concepts

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

Chain Rule
The chain rule is a fundamental technique in calculus for differentiating compositions of functions. When you have a function nested within another function, the chain rule helps you find the overall derivative.

Imagine you have a function \(y = f(g(x))\). To differentiate this, you need to follow two steps:
  • Differentiate the outer function \(f\) with respect to its inner function \(g(x)\).
  • Multiply by the derivative of the inner function \(g(x)\).

For example, in the given problem, we have \(y = 7(1+x)^{-\frac{1}{2}}\). The outer function here is \(u = 7u\) and the inner function is \(u = (1+x)^{-\frac{1}{2}}\). Using the chain rule, you differentiate the outer function first and then multiply by the derivative of the inner function. This method helps handle more complex functions efficiently.
Power Rule
The power rule is a basic rule used to find the derivative of functions of the form \(y = x^n\), where \(n\) is any real number. The power rule states that if \(y = x^n\), then the derivative of \(y\) with respect to \(x\) is \(y' = nx^{n-1}\).

In the exercise, the power rule plays a crucial role when rewriting \( \frac{7}{\sqrt{1+x}} \) as \(7(1+x)^{-\frac{1}{2}}\). By applying the power rule directly, the derivative of \(y = (1+x)^{-\frac{1}{2}}\) becomes
  • \

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