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Chapter 2: Problem 22

Find \(d y / d x\). $$ y=\sqrt[4]{2 x+1} $$

Short Answer

Expert verified
The derivative is \( \frac{dy}{dx} = \frac{1}{2}(2x + 1)^{-3/4} \).

Step by step solution

01

Rewrite the function

The function given is \( y = \sqrt[4]{2x + 1} \). We can rewrite this as a power function: \( y = (2x + 1)^{1/4} \). This will make it easier to apply the power rule for differentiation.
02

Differentiate using the power rule

The power rule states that if \( y = u^n \), then \( \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \). Using this rule, we have \( n = \frac{1}{4} \) and \( u = 2x + 1 \). So, \( \frac{dy}{dx} = \frac{1}{4}(2x + 1)^{-3/4} \cdot \frac{d}{dx}(2x + 1) \).
03

Differentiate the inner function

Now, find \( \frac{d}{dx}(2x + 1) \). The derivative of \( 2x \) is 2 and the derivative of 1 is 0, so \( \frac{d}{dx}(2x + 1) = 2 \).
04

Simplify the expression

Substitute \( \frac{d}{dx}(2x + 1) = 2 \) back into the derivative expression to get \( \frac{dy}{dx} = \frac{1}{4}(2x + 1)^{-3/4} \cdot 2 \). Simplify to \( \frac{dy}{dx} = \frac{1}{2}(2x + 1)^{-3/4} \).

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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 fundamental tool in calculus, particularly when we are working with derivatives of functions in the form of a power. This rule states that for a function expressed as a power of another function, such as \( y = u^n \), the derivative \( \frac{dy}{dx} \) is obtained by multiplying the power \( n \) by the original function \( u \), raised to the power \( n-1 \), and then multiplying by the derivative of \( u \).
  • For example, if \( y = (2x + 1)^{1/4} \), the power rule helps us differentiate this by applying \( n = 1/4 \) and having \( u = 2x + 1 \).
  • The process involves calculating the derivative with respect to \( x \), which is \( \frac{dy}{dx} = \frac{1}{4}(2x + 1)^{-3/4} \cdot \frac{d}{dx}(2x + 1) \).
The power rule simplifies this complex operation, making differentiation straightforward once the function is appropriately rewritten.
Differentiation
Differentiation is a process in calculus that allows us to determine how a function changes as its input changes. It's essentially about finding the rate of change or the slope of a curve at any given point.
  • For functions like \( y = (2x + 1)^{1/4} \), differentiation involves using certain rules, such as the power rule, to systematically find \( \frac{dy}{dx} \).
  • At the core of differentiation is understanding how incremental changes in \( x \) influence changes in \( y \).
  • In this instance, after using the power rule, we further differentiate the inner function \( 2x + 1 \) to get a derivative of 2.
By combining these results, we can simplify the derivative to obtain \( \frac{dy}{dx} = \frac{1}{2}(2x + 1)^{-3/4} \), which reveals how the function \( y = (2x + 1)^{1/4} \) changes with \( x \).
Function Transformation
Function transformation involves changing the form of a function to make it easier to work with, especially when calculating derivatives. In our example, the original function was \( y = \sqrt[4]{2x + 1} \) and we transformed it to \( y = (2x + 1)^{1/4} \).
  • This transformation process is crucial because it simplifies functions, allowing us to apply rules like the power rule more directly.
  • When dealing with composite functions, such as those involving radicals or exponential powers, rewriting them into power form helps streamline the differentiation process.
Transforming functions efficiently can often make the difference between a difficult and manageable task, aiding in better understanding not only of derivatives but also the behavior of functions themselves.

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