find the derivatives of the given functions.
g(t)=sqrt t(1+t)/t^2
the sqrt is just over t.
To find the derivative of the given function, we can use the quotient rule and the chain rule.
Let's break down the function first:
g(t) = sqrt(t * (1 + t))/t^2
Step 1: Simplify the function
g(t) = sqrt(t^2 + t)/t^2
Step 2: Apply the quotient rule
If we have a function of the form f(t) = p(t)/q(t), where p(t) and q(t) are both differentiable functions, then the derivative of f(t) with respect to t can be calculated using the quotient rule:
f'(t) = (p'(t) * q(t) - p(t) * q'(t))/(q(t))^2
In our case,
p(t) = sqrt(t^2 + t)
q(t) = t^2
Step 3: Calculate the derivatives of p(t) and q(t)
To differentiate p(t), we need to use the chain rule since we have a composition of functions. The chain rule states that if we have a function of the form f(g(t)), the derivative with respect to t is given by f'(g(t)) * g'(t).
In our case,
Let u = t^2 + t
So, p(t) = sqrt(u)
Using the chain rule, we find that p'(t) = (1/2) * (u)^(-1/2) * (2t + 1) = (t + 1)/sqrt(t^2 + t)
To differentiate q(t), we can use the power rule for derivatives, which states that if f(t) = t^n, then f'(t) = n * t^(n-1).
In our case, q(t) = t^2, so q'(t) = 2t.
Step 4: Plug in the values into the quotient rule formula
Now we can substitute the values into the quotient rule formula:
g'(t) = [(p'(t) * q(t)) - (p(t) * q'(t))]/(q(t))^2
= [(t + 1)/sqrt(t^2 + t) * t^2 - sqrt(t^2 + t) * 2t]/(t^2)^2
= [(t + 1)t^2 - 2t(sqrt(t^2 + t))]/t^4
= (t^3 + t^2 - 2t(sqrt(t^2 + t)))/t^4
Simplifying further, we get:
g'(t) = (t^3 + t^2 - 2t*sqrt(t^2 + t))/t^4
Therefore, the derivative of g(t) = sqrt(t * (1 + t))/t^2 is g'(t) = (t^3 + t^2 - 2t*sqrt(t^2 + t))/t^4.