Question

attempt 6: 5 attempts remaining. given ( y = \frac{x^5 - 1}{x^5 + 1} ), find ( \frac{dy}{dx} ). ( \frac{dy}{dx} = ) video example: solving a similar problem

Explanation:

Step1: Identify the quotient rule

The function \( y = \frac{u}{v} \) where \( u = x^5 - 1 \) and \( v = x^5 + 1 \). The quotient rule states that \( \frac{dy}{dx}=\frac{u'v - uv'}{v^2} \).

Step2: Find \( u' \) and \( v' \)

Differentiate \( u = x^5 - 1 \) with respect to \( x \): \( u' = 5x^4 \) (using the power rule \( \frac{d}{dx}(x^n)=nx^{n - 1} \)).
Differentiate \( v = x^5 + 1 \) with respect to \( x \): \( v' = 5x^4 \).

Step3: Substitute into the quotient rule

Substitute \( u, v, u', v' \) into \( \frac{u'v - uv'}{v^2} \):
\[

$$\begin{align*} \frac{dy}{dx}&=\frac{(5x^4)(x^5 + 1)-(x^5 - 1)(5x^4)}{(x^5 + 1)^2}\\ &=\frac{5x^9 + 5x^4 - 5x^9 + 5x^4}{(x^5 + 1)^2}\\ &=\frac{10x^4}{(x^5 + 1)^2} \end{align*}$$

\]

Answer:

\( \frac{10x^4}{(x^5 + 1)^2} \)