Please can someone help. I need to use the composite rule to differentiate the function f(x)=(x^2-6x+23)^3/2. Any help would be greatly appreciated.

I don't know what name they give the rule these days, but the way I do it would get you ...

f'(x) = (3/2)(2x-6)(x^2 - 6x + 23)^(1/2) in one step.
and
= 3(x-3)√(x^2 - 6x + 23)

Of course! I'd be happy to help you differentiate the function f(x) = (x^2 - 6x + 23)^(3/2) using the composite rule. Here's how you can do it step by step:

Step 1: Identify the inner and outer functions. In this case, the inner function is (x^2 - 6x + 23) and the outer function is ( )^(3/2).

Step 2: Differentiate the outer function. To differentiate ( )^(3/2), we'll use the power rule for differentiation. The power rule states that if the function is of the form f(x) = u^n, where n is a constant, then its derivative is f'(x) = n*u^(n-1)*u'. In this case, u is our inner function (x^2 - 6x + 23), and n is 3/2. Applying the power rule, the derivative of (x^2 - 6x + 23)^(3/2) is (3/2)*(x^2 - 6x + 23)^(1/2) * (d/dx)(x^2 - 6x + 23).

Step 3: Differentiate the inner function. To differentiate (x^2 - 6x + 23), we'll use the sum and constant multiple rules. The sum rule states that if f(x) = u + v, then f'(x) = u' + v'. In this case, u is x^2, and v is -6x + 23. Applying the sum rule, the derivative of (x^2 - 6x + 23) is (d/dx)(x^2) + (d/dx)(-6x + 23). The derivative of x^2 is 2x, and the derivative of -6x + 23 is -6.

Step 4: Combine the derivatives. Now we have the derivative of the outer function (3/2)*(x^2 - 6x + 23)^(1/2) * (2x - 6). Multiplying this out, we get (3x - 9)*(x^2 - 6x + 23)^(1/2).

And that's it! You have successfully differentiated the function f(x) = (x^2 - 6x + 23)^(3/2) using the composite rule. The derivative is (3x - 9)*(x^2 - 6x + 23)^(1/2).