f(x)=(�ãx+9x)(x^3/2-x)
Evaluate f '(x) at the value x = 4.
f '(4) = ??
To evaluate the derivative of f(x) at x = 4, we will find the derivative of the function f(x) and then substitute x = 4 into the derivative expression.
Let's start by finding the derivative of f(x). To do this, we will apply the product rule and the power rule.
The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by (u(x)v'(x) + u'(x)v(x)).
In our case, u(x) = (x + 9x) and v(x) = (x^(3/2) - x).
Applying the product rule, we have:
f '(x) = (u(x)v'(x) + u'(x)v(x))
= ((x + 9x)(d/dx(x^(3/2) - x)) + (d/dx(x + 9x))(x^(3/2) - x))
Next, we find the derivative of the function v(x) = (x^(3/2) - x).
Using the power rule, the derivative of x^(3/2) is (3/2)x^(3/2 - 1) = (3/2)x^(1/2).
The derivative of -x is -1.
Thus, v'(x) = (3/2)x^(1/2) - 1.
Substituting this into the expression for f '(x), we get:
f '(x) = ((x + 9x)((3/2)x^(1/2) - 1) + (1 + 9))(x^(3/2) - x))
Now, we substitute x = 4 into this expression to find f '(4):
f '(4) = ((4 + 9(4))((3/2)(4)^(1/2) - 1) + (1 + 9))(4^(3/2) - 4))
Simplifying this expression will give us the value of f '(4).