(dy)/(dx)(x)/((3x-2)^(9))

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Identify u and v: We need to find the derivative of the function (x)/((3x-2)^(9)) with respect to x. This is a quotient, so we will use the quotient rule which states that (d/dx)(u/v) = (v(u') - u(v')) / v^2, where u is the numerator and v is the denominator.Find derivative of u: Let's identify u and v for our function. Here, u = x and v = (3x-2)^9. We will need to find the derivatives of u and v, which are u' and v' respectively.Find derivative of v: First, we find the derivative of u with respect to x. Since u = x, the derivative u' is simply 1 because the derivative of x with respect to x is 1. u' = (d/dx)(x) = 1Apply quotient rule: Next, we find the derivative of v with respect to x. Since v = (3x-2)^9, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is f(t) = t^9 and the inner function is g(x) = 3x-2. v' = (d/dx)((3x-2)^9) = 9(3x-2)^8 * (d/dx)(3x-2) = 9(3x-2)^8 * 3Simplify expression: Now we can apply the quotient rule. We have u' = 1 and v' = 9(3x-2)^8 * 3. Plugging these into the quotient rule formula, we get: (dy/dx) = ((3x-2)^9 * 1 - x * 9(3x-2)^8 * 3) / (3x-2)^18Factor out common term: Simplify the expression by distributing and combining like terms: (dy/dx) = ((3x-2)^9 - 27x(3x-2)^8) / (3x-2)^18Cancel out common term: We can factor out a (3x-2)^8 from the numerator to simplify the expression further: (dy/dx) = (3x-2)^8 * (1 - 27x) / (3x-2)^18Final derivative: Now we can cancel out (3x-2)^8 from the numerator and denominator: (dy/dx) = (1 - 27x) / (3x-2)^10We have found the derivative of the function (x)/((3x-2)^(9)) with respect to x, which is: (dy/dx) = (1 - 27x) / (3x-2)^10

$\frac{dy}{dx}\left(x\right)\left(\frac{1}{\left(3x-2\right)^{9}}\right)$

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$\frac{dy}{dx}\left(x\right)\left(\frac{1}{\left(3x-2\right)^{9}}\right)$