Questions: Find the derivative of the function. y=(3x^2-x)/(9x-2)^4 dy/dx=

Transcript text: Find the derivative of the function. \[ \begin{array}{l} y=\frac{3 x^{2}-x}{(9 x-2)^{4}} \\ \frac{d y}{d x}=\square \end{array} \]

Solution

Solution Steps

To find the derivative of the given function, we will use the quotient rule. The quotient rule states that if you have a function \( y = \frac{u(x)}{v(x)} \), then its derivative is given by \( \frac{d y}{d x} = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = 3x^2 - x \) and \( v(x) = (9x - 2)^4 \). We will first find the derivatives \( u'(x) \) and \( v'(x) \), and then apply the quotient rule.

Step 1: Define the Functions

We start with the function given by \[ y = \frac{3x^2 - x}{(9x - 2)^4} \] where we define \( u = 3x^2 - x \) and \( v = (9x - 2)^4 \).

Step 2: Calculate the Derivatives

Next, we compute the derivatives of \( u \) and \( v \): \[ u' = \frac{d}{dx}(3x^2 - x) = 6x - 1 \] \[ v' = \frac{d}{dx}((9x - 2)^4) = 36(9x - 2)^3 \]

Step 3: Apply the Quotient Rule

Using the quotient rule, we find the derivative \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{u'v - uv'}{v^2} \] Substituting the values we calculated: \[ \frac{dy}{dx} = \frac{(6x - 1)(9x - 2)^4 - (3x^2 - x)(36(9x - 2)^3)}{(9x - 2)^8} \]