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Chapter 3: Problem 62

Determine if the derivative rules from this section apply. If they do, find the derivative. If they don't apply, indicate why. $$y=(x+3)^{1 / 2}$$

Short Answer

Expert verified
Derivative rules apply; derivative is \( \frac{1}{2\sqrt{x+3}} \).

Step by step solution

01

Identify the Function Type

The given function is \[ y = (x + 3)^{1/2} \]This is a composite function where the outer function is \(f(u) = u^{1/2}\) and the inner function is \(u = x + 3\).
02

Confirm Applicability of Chain Rule

Since the given function is a composition of functions, and both functions are differentiable, the chain rule can be applied. The chain rule is used to differentiate a composite function \(f(g(x))\).
03

Differentiate the Outer Function

The derivative of the outer function \(f(u) = u^{1/2}\) with respect to \(u\) is \[ f'(u) = \frac{1}{2}u^{-1/2} \].
04

Differentiate the Inner Function

The derivative of the inner function \(u = x + 3\) with respect to \(x\) is \[ \frac{d}{dx}[x + 3] = 1 \].
05

Apply the Chain Rule

Using the chain rule, the derivative of \(y = (x + 3)^{1/2}\) is\[ \frac{dy}{dx} = f'(u) \cdot \frac{du}{dx} = \frac{1}{2}(x+3)^{-1/2} \cdot 1 \].
06

Simplify the Derivative Expression

The simplified form of the derivative is \[ \frac{dy}{dx} = \frac{1}{2\sqrt{x+3}} \].

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Key Concepts

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

Composite Function
A composite function is essentially a function of a function. It occurs when one function is nested within another. In our example, we have \[ y = (x + 3)^{1/2} \], where the expression involves two functions: an outer function and an inner function.
The outer function is \( f(u) = u^{1/2} \), a square root transformation. Meanwhile, the inner function is \( g(x) = x + 3 \). The composite nature of this function is why you must differentiate with care.
  • Composite functions often require special rules (like the chain rule) for differentiation.
  • Identify the "outer" and "inner" functions to apply the chain rule effectively.
Recognizing composite functions helps in correctly applying differentiation techniques.
Derivative
The concept of a derivative revolves around the idea of finding the rate at which a function changes with respect to a variable. Simply put, it's a measure of how a function's output changes as the input changes.
For the function \( y = (x + 3)^{1/2} \), we are interested in finding this rate of change with respect to \( x \). To do this, we differentiate the outer function and the inner function:
  • The derivative of the outer function \( f(u) = u^{1/2} \) with respect to \( u \) is \( f'(u) = \frac{1}{2}u^{-1/2} \).
  • For the inner function \( u = x + 3 \), the derivative with respect to \( x \) is simply \( 1 \) since it's a linear function.
This two-step differentiation process is essential when dealing with composite functions.
Differentiable Functions
To apply concepts like the derivative or the chain rule, the functions involved must be differentiable. A function is differentiable at a point if its derivative exists at that specific point.
In our exercise, both the outer function \( f(u) = u^{1/2} \) and the inner function \( u = x + 3 \) are differentiable. This means:
  • The derivative of \( f(u) = u^{1/2} \) exists because it's a type of power function, which is differentiable for all its domain.
  • The inner linear function \( u = x + 3 \) is differentiable everywhere since linear functions have derivatives that exist everywhere along their graph.
If either function wasn't differentiable, the chain rule would not be applicable. Ensuring differentiability is crucial in calculus to utilize the power of derivatives.

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Most popular questions from this chapter

Find the limit of the function as \(x \rightarrow \infty\). $$\frac{\sinh \left(x^{2}\right)}{\cosh \left(x^{2}\right)}$$

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