If I derive the following,
d/dt((a)S(t^2) of (s)^(1/2)ds
with S being integral notation
(a) being at the bottom of the notation and being a constant
(t^2) being at the top of the notation
..I first apply the fundamental theorem of calculus to get..
(t^2)^(1/2) (2t)
simplified to 2(t^2)
Is this correct?
To solve this problem, let's break it down step by step.
First, we have the expression: d/dt((a)S(t^2) of (s)^(1/2)ds
To begin, we can rewrite the expression using the fundamental theorem of calculus, which states that the derivative of an integral with respect to a variable is just the integrand evaluated at the upper limit of integration multiplied by the derivative of the upper limit.
So, applying the fundamental theorem of calculus, we get:
d/dt((a)S(t^2) of (s)^(1/2)ds) = S(t^2)^(1/2) * (d/dt(t^2))
Now, let's differentiate (d/dt(t^2)):
Since t^2 is a simple polynomial, the derivative of t^2 with respect to t is simply 2t.
So, we can replace (d/dt(t^2)) with 2t:
d/dt((a)S(t^2) of (s)^(1/2)ds) = S(t^2)^(1/2) * 2t
At this point, we can simplify the expression:
S(t^2)^(1/2) can be written as S(t^2)^(1/2) = S(t^(2*(1/2))) = S(t^1) = S(t)
Therefore, the simplified expression becomes:
d/dt((a)S(t^2) of (s)^(1/2)ds) = 2t * S(t)
So, to answer your question, the correct derivative is not 2(t^2), but rather 2t * S(t).
I hope this explanation helps! Let me know if you have any further questions.