How would I find the derivative of (8x^3)/sqrt(3x-2)?
I would write it as
y = (8x^3)(3x-2)^(-1/2) and use the product rule
dy/dx = (8x^3)(-1/2)(3x-2)^(-3/2) (3) + 24x^2 (3x-2)^(1/2)
I will let you clean it up a bit
To find the derivative of the given function, you can use the quotient rule. The quotient rule states that for a function in the form of f(x) = g(x) / h(x), where g(x) and h(x) are both functions of x, the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
Let's break it down step by step:
Step 1: Identify g(x) and h(x) in the function given.
g(x) = 8x^3
h(x) = sqrt(3x - 2)
Step 2: Find the derivatives of g(x) and h(x).
g'(x) = 24x^2
h'(x) = (1 / 2 * sqrt(3x - 2)) * 3
Step 3: Plug the values into the quotient rule formula.
f'(x) = (24x^2 * sqrt(3x - 2) - 8x^3 * (1 / 2 * sqrt(3x - 2)) * 3) / (sqrt(3x - 2))^2
Step 4: Simplify the expression.
f'(x) = (24x^2 * sqrt(3x - 2) - 12x^3) / (3x - 2)
Therefore, the derivative of (8x^3) / sqrt(3x - 2) is (24x^2 * sqrt(3x - 2) - 12x^3) / (3x - 2).