Question

Find \(\frac{d y}{d x}\). $$y=\frac{x}{\sqrt[5]{x}+x}$$

Short answer

Answer: The derivative of the function y = (x) / (∛x + x) is dy/dx = (x^(1/5) - (1/5)x^(6/5)) / (x^(1/5) + x)^2.

Step-by-step solution

Find the derivative of f(x)

To find the derivative of $$f(x)=x$$, we can use the power rule, which states that:$$\frac{d}{d x}(x^n)=n x^{n-1}$$In this case, $$n=1$$, so we get the derivative $$f'(x)=1$$.

Find the derivative of g(x)

To find the derivative of $$g(x)=\sqrt[5]{x}+x$$, we will treat it as a sum of two functions and use the sum rule: The sum rule tells us that:$$\frac{d}{d x}(u+v)=\frac{d u}{d x}+\frac{d v}{d x}$$where $$u=\sqrt[5]{x}$$ and $$v=x$$. To find the derivative of $$u$$, we can rewrite it as:$$u=x^{\frac{1}{5}}$$and then use the power rule again. In this case, $$n=\frac{1}{5}$$, we get the derivative:$$\frac{d}{d x}(u)=\frac{1}{5}x^{-\frac{4}{5}}$$ To find the derivative of $$v$$, as we have already calculated in Step 1, we get:$$\frac{d}{d x}(v)=1$$ Now, we can use the sum rule to find the derivative of $$g(x)$$:$$g'(x)=\frac{1}{5}x^{-\frac{4}{5}}+1$$

Apply the quotient rule

Now that we have the derivatives of both functions, we can apply the quotient rule to find the derivative of $$y$$: $$\frac{d y}{d x}=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$$ Plugging in the derived expressions and the original functions, we get: $$\frac{d y}{d x}=\frac{(1)(\sqrt[5]{x}+x)-(x)(\frac{1}{5}x^{-\frac{4}{5}}+1)}{(\sqrt[5]{x}+x)^2}$$

Simplify the expression

Now, we will simplify the expression for the derivative: $$\frac{d y}{d x}=\frac{\sqrt[5]{x}+x-\frac{1}{5}x^{\frac{1}{5}}x-\cancel{x}}{(\sqrt[5]{x}+x)^2}$$ $$\frac{d y}{d x}=\frac{\sqrt[5]{x}-\frac{1}{5}x^{\frac{6}{5}}}{(\sqrt[5]{x}+x)^2}$$ So, the derivative of the given function is: $$\boxed{\frac{d y}{d x}=\frac{\sqrt[5]{x}-\frac{1}{5}x^{\frac{6}{5}}}{(\sqrt[5]{x}+x)^2}}$$

Key concepts

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. If you have \(y = \frac{f(x)}{g(x)}\), the quotient rule formula is: \ \frac{d}{dx}\left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \.

To apply the quotient rule, remember these key steps:

• First, differentiate the numerator function, which gives \(f'(x)\).
• Second, differentiate the denominator function, which gives \(g'(x)\).
• Plug these into the quotient rule formula, then simplify the expression.

In practical terms, this rule is useful whenever you are dealing with a ratio of two functions, much like fractions in algebra. Pay extra attention to applying it carefully. The rule helps in maintaining the integrity of derivative calculations involving divisions.
In our exercise, we applied the quotient rule to differentiate \(y = \frac{x}{\sqrt[5]{x} + x}\). By finding the derivatives \(f'(x)\) and \(g'(x)\), and plugging them into the formula, we were able to arrive at the final simplified derivative.

Power Rule

The power rule is a simple and widely used method to find derivatives of functions where the variable is raised to a power. It states that for any function \(f(x) = x^n\), the derivative is \ \frac{d}{dx}(x^n) = nx^{n-1} \.

Let’s break down how you apply this rule:

• Identify the power \(n\) in the function.
• Multiply the entire function by this power.
• Subtract one from the power to represent the new power of the function.

This rule is particularly useful when dealing with polynomial functions or any variable raised to a power. It simplifies the process of differentiation considerably, making it both quicker and efficient.
The power rule was used twice in this exercise: first to find the derivative of \(f(x) = x\), resulting in \(f'(x) = 1\), and next for the function \(u = x^{\frac{1}{5}}\) forming part of \(g(x) = \sqrt[5]{x} + x.\) This showcases its versatility.

Sum Rule

The sum rule helps you find the derivative of a function that is a sum of two or more functions. It states: \( \frac{d}{dx}(u+v) = \frac{du}{dx} + \frac{dv}{dx} \). It’s straightforward and vital for breaking down complex derivatives into manageable parts.

Here's how you can implement it:

• Separate the function into individual components \(u\) and \(v\).
• Differentiate each component independently.
• Add the results to find the overall derivative.

The sum rule is particularly helpful when faced with a function built from other, simpler functions that are added together. This modular approach also makes understanding the derivative process clearer.
In this exercise, the sum rule was employed to differentiate the function \(g(x) = \sqrt[5]{x} + x\) by treating it as a sum of \(u = x^{\frac{1}{5}}\) and \(v = x\). Each derivative was calculated separately before being summed up, leading to a neat and organized approach to solving the problem.