Question
Example 13 Given \( y=\frac{2 x^{6}}{4-x} \), find \( \frac{d y}{d x} \)
Ask by Mathis Norton. in Singapore
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The derivative of \( y = \frac{2x^{6}}{4 - x} \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{2x^{5} (24 - 5x)}{(4 - x)^{2}} \).
Solution
To find the derivative \( \frac{dy}{dx} \) of the function \( y = \frac{2x^{6}}{4 - x} \), we can use the **quotient rule**. The quotient rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then its derivative is:
\[
\frac{dy}{dx} = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
**Step-by-Step Solution:**
1. **Identify \( u(x) \) and \( v(x) \):**
\[
u(x) = 2x^6 \quad \text{and} \quad v(x) = 4 - x
\]
2. **Find the derivatives \( u'(x) \) and \( v'(x) \):**
\[
u'(x) = \frac{d}{dx}(2x^6) = 12x^5
\]
\[
v'(x) = \frac{d}{dx}(4 - x) = -1
\]
3. **Apply the quotient rule:**
\[
\frac{dy}{dx} = \frac{12x^5(4 - x) - 2x^6(-1)}{(4 - x)^2}
\]
4. **Simplify the numerator:**
\[
= \frac{12x^5(4 - x) + 2x^6}{(4 - x)^2}
\]
\[
= \frac{48x^5 - 12x^6 + 2x^6}{(4 - x)^2}
\]
\[
= \frac{48x^5 - 10x^6}{(4 - x)^2}
\]
5. **Factor out the common terms in the numerator:**
\[
= \frac{2x^5(24 - 5x)}{(4 - x)^2}
\]
**Final Answer:**
\[
\frac{dy}{dx} = \frac{2x^{5} \left(24 - 5x\right)}{(4 - x)^{2}}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution Mind Expander
To differentiate the function \( y = \frac{2 x^{6}}{4-x} \), we can use the quotient rule, which states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( \frac{dy}{dx} \) is given by \( \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} \). Here, let: - \( u = 2x^6 \) with \( \frac{du}{dx} = 12x^5 \) - \( v = 4 - x \) with \( \frac{dv}{dx} = -1 \) Now applying the quotient rule: \[ \frac{dy}{dx} = \frac{(4-x)(12x^5) - (2x^6)(-1)}{(4-x)^2} \] Calculating the numerator: \[ (4-x)(12x^5) + 2x^6 = 48x^5 - 12x^6 + 2x^6 = 48x^5 - 10x^6 \] Thus, we have: \[ \frac{dy}{dx} = \frac{48x^5 - 10x^6}{(4-x)^2} \] This is the derivative of \( y \) with respect to \( x \).
