Differentiate and simplify the following using the product rule and simplify completely: y = (√x)(1/x)
Nevermind, I solved it. The final result is -x/(2x^2√x), which can be simplified further into: -1/2√(x^3)
(√x)(1/x) = 1/√x = x^(-1/2)
so you are correct. The derivative is -1/2 x^(-3/2)
But I guess you simplified the result of the product rule:
1/2 (1/√x)*(1/x) + (√x)(-1/x^2)
To differentiate the function y = (√x)(1/x) using the product rule, we need to differentiate both terms separately and then combine the results.
Let's start by finding the derivative of the first term, (√x). To do this, we can use the power rule. The power rule states that if we have a function of the form f(x) = x^n, then its derivative is f'(x) = nx^(n-1).
In this case, (√x) can be also written as x^(1/2), so its derivative is:
d/dx(√x) = d/dx(x^(1/2)) = (1/2)x^(-1/2).
Now, let's find the derivative of the second term, (1/x). The derivative of 1/x can be obtained using the quotient rule, but since the numerator is 1, we can simplify it by using the power rule. If we have f(x) = 1/x, then its derivative is f'(x) = -1/x^2.
Next, we apply the product rule, which states that if we have a function of the form f(x) = u(x)v(x), where u(x) and v(x) are both differentiable functions of x, then its derivative is f'(x) = u'(x)v(x) + u(x)v'(x).
Applying the product rule to our function y = (√x)(1/x), we have:
dy/dx = (1/2)x^(-1/2) * (1/x) + (√x) * (-1/x^2).
Simplifying this expression, we obtain:
dy/dx = (1/2x^(1/2)) * (1/x) - (√x/x^2).
Now, let's simplify this further:
dy/dx = (1/(2x^(1/2)x)) - (√x/x^2) = 1/(2x) - (√x)/x^2.
To completely simplify this expression, we can find a common denominator and combine the fractions:
dy/dx = (x - 2√x)/2x^2.
Therefore, the simplified expression for the derivative of y = (√x)(1/x) using the product rule is dy/dx = (x - 2√x)/2x^2.