Find the derivative of the functions:
1) f(x)=x(1-(2)/(x+1))
2) f(s)=(s^2-2s+5)/(x^(1/2))
To find the derivative of a function, we can use the rules of differentiation.
1) f(x) = x(1 - 2/(x + 1))
To differentiate this function, we can use the product rule, which states that the derivative of a product of two functions is equal to the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Let's break down our function into two parts: f(x) = g(x) * h(x), where g(x) = x and h(x) = (1 - 2/(x + 1)).
Now, let's find the derivatives of g(x) and h(x):
The derivative of g(x) = x is simply 1.
To find the derivative of h(x), we need to use the quotient rule. The quotient rule states that the derivative of a quotient of two functions is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
Let's find the derivatives of the numerator and denominator of h(x):
Derivative of numerator = 1, since it is a constant.
Derivative of denominator = 1/(x + 1)^2, using the chain rule.
Now, applying the quotient rule, we have:
h'(x) = [(1)(x + 1)^2 - (1 - 2/(x + 1))(1)] / (x + 1)^2
= [(x + 1)^2 - (x + 1)(1 - 2/(x + 1))] / (x + 1)^2
= [(x + 1)^2 - (x + 1) + 2] / (x + 1)^2
= [x^2 + 2x + 1 - x - 1 + 2] / (x + 1)^2
= (x^2 + x + 2) / (x + 1)^2
Now, applying the product rule, we have:
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= 1 * (1 - 2/(x + 1)) + x * (x^2 + x + 2) / (x + 1)^2
= 1 - 2/(x + 1) + x * (x^2 + x + 2) / (x + 1)^2
Therefore, the derivative of f(x) = x(1 - 2/(x + 1)) is:
f'(x) = 1 - 2/(x + 1) + x * (x^2 + x + 2) / (x + 1)^2
2) f(s) = (s^2 - 2s + 5) / (x^(1/2))
To differentiate this function, we need to use the quotient rule again.
Let's break down our function into two parts: f(s) = p(s) / q(s), where p(s) = (s^2 - 2s + 5) and q(s) = (s^(1/2)).
Now, let's find the derivatives of p(s) and q(s):
The derivative of p(s) = 2s - 2, using the power rule and the constant rule.
The derivative of q(s) = (1/2) * s^(-1/2), using the power rule and the constant rule.
Applying the quotient rule, we have:
f'(s) = [q(s) * p'(s) - p(s) * q'(s)] / (q(s))^2
= [s^(1/2) * (2s - 2) - (s^2 - 2s + 5) * (1/2) * s^(-1/2)] / (s^(1/2))^2
= [2s^(3/2) - 2s^(1/2) - (1/2) * (s^2 - 2s + 5) * s^(-1/2)] / s
Simplifying further, we have:
f'(s) = 2s^(3/2) - 2s^(1/2) - (1/2) * (s^2 - 2s + 5) / s^(1/2)
Therefore, the derivative of f(s) = (s^2 - 2s + 5) / (s^(1/2)) is:
f'(s) = 2s^(3/2) - 2s^(1/2) - (1/2) * (s^2 - 2s + 5) / s^(1/2)