(dy)/(dx)(x)/((3x-2)^(6))

Question

Bytelearn · Accepted Answer

Identify u and v: We need to find the derivative of the function (x)/((3x-2)^(6)) 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 u' and v': Let's identify u and v for our function. Here, u = x and v = (3x-2)^6. We will need to find the derivatives of u and v, denoted as u' and v' respectively.Derivative of u: First, we find the derivative of u with respect to x. Since u = x, the derivative u' = (d/dx)(x) = 1.Derivative of v: Next, we find the derivative of v with respect to x. Since v = (3x-2)^6, 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. So, v' = (d/dx)((3x-2)^6) = 6*(3x-2)^5 * (d/dx)(3x-2) = 6*(3x-2)^5 * 3.Apply quotient rule: Now we apply the quotient rule. The derivative of our function with respect to x is given by: (dy/dx) = ((3x-2)^6 * (d/dx)(x) - x * (d/dx)((3x-2)^6)) / ((3x-2)^6)^2 Substituting the derivatives we found, we get: (dy/dx) = ((3x-2)^6 * 1 - x * 6*(3x-2)^5 * 3) / ((3x-2)^12)Simplify expression: Simplify the expression by distributing and combining like terms: (dy/dx) = ((3x-2)^6 - 18x*(3x-2)^5) / ((3x-2)^12)Factor out: We can factor out a (3x-2)^5 from the numerator to simplify the expression further: (dy/dx) = (3x-2)^5 * (1 - 18x) / ((3x-2)^12)Cancel out: Now we can cancel out (3x-2)^5 from the numerator and denominator: (dy/dx) = (1 - 18x) / ((3x-2)^7)Final derivative: We have found the derivative of the function (x)/((3x-2)^(6)) with respect to x, which is (1 - 18x) / ((3x-2)^7).

$\frac{dy}{dx}\left(x\right)\left(\frac{1}{\left(3x-2\right)^{6}}\right)$

Full solution

More problems from Find derivatives of using multiple formulae

Related topics

dydx(x)(1(3x−2)6)\frac{dy}{dx}\left(x\right)\left(\frac{1}{\left(3x-2\right)^{6}}\right)dxdy​(x)((3x−2)61​)

$\frac{dy}{dx}\left(x\right)\left(\frac{1}{\left(3x-2\right)^{6}}\right)$