If f(x) = sqrt x + 3/sqrt x, then f’(4) =
since √x = x^(1/2)
f(x) = x^(1/2) + 3x^(-1/2)
f'(x) = 1/2 x^(-1/2) - 3/2 x^(-3/2)
and plugging in x=4, that is
f'(4) = (1/2)(1/2) - (3/2)(1/8) = 1/16
To find the derivative of f(x), we need to apply the quotient rule. The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is (u'(x)v(x) - u(x)v'(x)) / (v(x))^2.
In this case, u(x) = sqrt(x) + 3 and v(x) = sqrt(x). Let's calculate their derivatives.
First, let's find u'(x):
u'(x) = d/dx (sqrt(x) + 3)
= d/dx (x^(1/2) + 3)
= (1/2)x^(-1/2)
Next, let's find v'(x):
v'(x) = d/dx (sqrt(x))
= d/dx (x^(1/2))
= (1/2)x^(-1/2)
Now, we can apply the quotient rule:
f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2
= ((1/2)x^(-1/2) * sqrt(x) - (sqrt(x) + 3) * (1/2)x^(-1/2)) / (sqrt(x))^2
= ((sqrt(x)/2) - (sqrt(x) + 3)/2) / x
= (sqrt(x) - sqrt(x) - 3)/2) / x
= (-3/2) / x
= -3 / (2x)
Now, let's find f'(4):
f'(4) = -3 / (2*4)
= -3 / 8
So, f’(4) = -3/8.
To find f'(4), the derivative of the function f(x), we can follow these steps:
Step 1: Begin with the given function:
f(x) = √x + 3/√x
Step 2: Start by differentiating the first term, which is √x.
The derivative of √x can be found using the Power rule, which states that if we have a function of the form f(x) = x^n, where n is a constant, then the derivative of f(x) is f'(x) = n * x^(n-1).
Applying the Power rule to √x, we have:
d(√x)/dx = (1/2) * x^(-1/2)
Step 3: Now let's differentiate the second term, which is 3/√x.
The derivative of 3/√x can be found using the Quotient rule, which states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) is f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / [h(x)]^2.
Applying the Quotient rule to 3/√x, we have:
d(3/√x)/dx = [(0 * √x) - (3 * (1/2) * x^(-1/2))] / (√x)^2
= (-3/2) * x^(-1/2) / x
Step 4: Simplify the derivative obtained from steps 2 and 3.
d(√x + 3/√x)/dx = (1/2) * x^(-1/2) + (-3/2) * x^(-1/2) / x
= (1/2) * x^(-1/2) - (3/2) * x^(-1) / x
= (1/2) * x^(-1/2) - (3/2) / x^(3/2)
= [x - 3] / (2x^(3/2))
Step 5: Evaluate f'(4) by plugging x = 4 into the derivative.
f'(4) = [4 - 3] / [2(4)^(3/2)]
= 1 / [2(8)]
= 1/16
So, f'(4) = 1/16.