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Chapter 2: Problem 7

Differentiate. $$ y=\sqrt{1-3 x} $$

Short Answer

Expert verified
\( \frac{dy}{dx} = -\frac{3}{2 \sqrt{1-3x}} \)

Step by step solution

01

- Identify the Outer Function

The function given is in the form of a composite function. Identify the outer function. Here, the outer function is the square root, which can be expressed as \( y = (1-3x)^{1/2} \).
02

- Identify the Inner Function

Next, determine the inner function contained within the outer function. Here, the inner function is \( u = 1 - 3x \).
03

- Differentiate the Outer Function

Differentiate the outer function with respect to the inner function \( u \). The outer function is \( y = u^{1/2} \). Using the power rule, the derivative is \[ \frac{dy}{du} = \frac{1}{2} u^{-1/2} = \frac{1}{2 \sqrt{u}} \].
04

- Differentiate the Inner Function

Differentiate the inner function \( u \) with respect to \( x \). That is, \[ \frac{du}{dx} = \frac{d}{dx}(1 - 3x) = -3 \].
05

- Apply the Chain Rule

Apply the chain rule which states that \[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]. Substituting the values, we get \[ \frac{dy}{dx} = \frac{1}{2 \sqrt{u}} \cdot (-3) = -\frac{3}{2 \sqrt{u}} \].
06

- Substitute Back the Inner Function

Finally, substitute back the inner function \( u = 1-3x \) into the expression obtained in the previous step. Thus, \[ \frac{dy}{dx} = -\frac{3}{2 \sqrt{1-3x}} \].

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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 a basic formula in calculus for differentiating functions of the form \( x^n \), where \( n \) is any real number. According to the power rule:
\( \frac{d}{dx} (x^n) = nx^{n-1} \)
This rule makes it straightforward to find the derivative of polynomial functions and is often a component of more complex differentiation problems. In our exercise, we use the power rule when differentiating the outer function (the square root), which we rewrite in exponential form as: \( (1-3x)^{1/2} \). Applying the power rule, we get:
\[ \frac{d}{du} u^{1/2} = \frac{1}{2} u^{-1/2} \]
Simplifying this expression provides the derivative needed for our chain rule application.

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