find f'(x)
f(x)=(3x^2-4)^5 / √2x-1
To find the derivative of f(x), denoted as f'(x), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then the derivative f'(x) can be calculated using the formula:
f'(x) = (h(x)*g'(x) - g(x)*h'(x)) / (h(x))^2
In this case, our function is f(x) = (3x^2-4)^5 / √(2x-1), where g(x) = (3x^2-4)^5 and h(x) = √(2x-1).
Now, let's find the derivative step by step:
Step 1: Find g'(x), the derivative of g(x).
To compute g'(x), we can use the chain rule, which states that if we have a function in the form f(g(x)), then the derivative is given by f'(g(x)) * g'(x).
In this case, f(g(x)) = g(x)^5, so by the chain rule, the derivative is:
g'(x) = 5 * g(x)^4 * g'(x)
Step 2: Find h'(x), the derivative of h(x).
To find the derivative of √(2x-1), we can use the power rule. The power rule states that if we have a function in the form f(x) = x^n, then the derivative is given by f'(x) = n * x^(n-1).
In this case, n = 1/2, so we have:
h'(x) = (1/2) * (2x-1)^(1/2 - 1) * 2
Step 3: Calculate f'(x) using the quotient rule.
Using the quotient rule, we have:
f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2
= (√(2x-1) * 5 * g(x)^4 * g'(x) - g(x) * (1/2) * (2x-1)^(1/2 - 1) * 2) / (√(2x-1))^2
= (5 * √(2x-1) * (3x^2-4)^4 * g'(x) - g(x) * (2x-1)^(-1/2) * 2) / (2x-1)
Therefore, f'(x) = (5 * √(2x-1) * (3x^2-4)^4 * g'(x) - 2 * g(x) * (2x-1)^(-1/2)) / (2x-1) is the derivative of f(x).