Calculate d/dt when a=t^2, b=2 and sqrt(x+1)dx.

My answer turns out to be incorrect despite my checking it several times. I got -2/3t* sqrt(t^2+1)

Can someone explain to me what is wrong with my answer and how to go about obtaining it the right way? Thank you everyone. :-)

f(t) = ∫[2,t^2] √(x+1) dx

df/dt = √(t^2+1) (2t)

Recall that d/dt ∫[a,b(t)] f(x) dx = f(b(t)) db/dt

see wikipedia's article on differentiating under the integral sign.

Thank you for the equation and the useful article. :)

So this is what I get.

f(t) dt= 2(0)- [sqrt(t^2+1) * 2t]
f(t) dt= -sqrt(t^2+1) *2t

Is this correct? Thank you so much. :-)

To find the derivative of the given functions, let's start by differentiating each term separately.

For a = t^2, we can use the power rule for differentiation. According to the power rule, when differentiating t^k with respect to t, where k is a constant, the result is k*t^(k-1). Applying this rule, we have:

d/dt(a) = d/dt(t^2) = 2*t^(2-1) = 2t

So, the derivative of a with respect to t is 2t.

Now, let's move on to b = 2. Here, b is just a constant, and the derivative of a constant with respect to any variable is always zero. Therefore, d/dt(b) = 0.

Finally, for sqrt(x+1)dx, we need to use the chain rule for differentiation. The chain rule states that when differentiating a composite function, such as f(g(x)), the result is f'(g(x)) * g'(x). In this case, x is a variable, and the function is sqrt(x+1).

Let's break it down step by step:
1. Substitute x with t^2 (as given).
2. Differentiate sqrt(t^2 + 1) with respect to t.
3. Multiply the result by dt to indicate that we are differentiating with respect to t.

Applying the chain rule, we have:

d(sqrt(x+1)) = d(sqrt(t^2 + 1))
= (1/2)*(t^2 + 1)^(-1/2) * d(t^2 + 1)
= (1/2)*(t^2 + 1)^(-1/2) * (2t)
= t/(sqrt(t^2 + 1))

Multiplying the result by dx, we obtain:

(sqrt(x+1))dx = (sqrt(t^2 + 1))dt

Now, let's summarize the derivatives we found:

d/dt(a) = 2t
d/dt(b) = 0
(sqrt(x+1))dx = (sqrt(t^2 + 1))dt

Comparing this with your answer, -2/3t*sqrt(t^2+1), there seem to be two errors:
1. The constant multiplier is incorrect. It should be 1/2 instead of -2/3.
2. There is an absence of t in the expression. It should be t/(sqrt(t^2 + 1)) instead of -2/3t.

To obtain the correct answer, re-evaluate the problem using the correct derivatives:

d/dt(a) = 2t
d/dt(b) = 0
(sqrt(x+1))dx = (sqrt(t^2 + 1))dt

If you go through these steps again, you should arrive at the correct derivative.