Example 13 Given \( y=\frac{2 x^{6}}{4-x} \), find \( \frac{d y}{d x} \)

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 \( x \).