Find dy/dx and d2y/dx2 if y= definite integral sign where a= 1 and b= 3x

1/(t^2+t+1) dt

how do i even start. do i integrate and then plug in a and b? plz help.

dy/dx = 1/(x^2+x+1) * 3

can you explain how you got this. thanks

excuse me? Didn't we just go through this about differentiating under the integral sign?

but i don't get how you can just replace the variable t with x so easily. i don't get how you did that. and how does the 3 come into the picture for dy/dx.

In general (see the wikipedia article for proof),

∫[a(x),b(x)] f(t) dt
= f(b(x)) db/dx - f(a(x)) da/dx

Since we have a(x) = 1, da/dx = 0 and we are left with

∫[a(x),b(x)] f(t) dt
= f(b(x)) db/dx

b(x) = 3x, so db/dx = 3

So, oops. I made a mistake. It should be

dy/dx = 1/((3x)^2+3x+1) * 3

Good catch! :-)

Thank you very much! I really didn't understand the form but now I do. :)

Also, would d2y/dx2 be:

-9(6x+1)/((9x^2+3x+1)^2)

I just want to make sure that d2y/dx2 is asking for the second derivative?

haven't checked your work, but yes, they want y"

Oh okay thank you!

To find the first derivative, dy/dx, and the second derivative, d2y/dx2, of the given function, you need to apply the Fundamental Theorem of Calculus and the Chain Rule.

Let's start with finding dy/dx:

Step 1: Apply the Fundamental Theorem of Calculus to find the derivative of the integral with respect to x:

dy/dx = d/dx ∫[a to b] f(t) dt

Step 2: Since the limits of integration involve the variable x, we need to apply the Chain Rule to differentiate the integrand f(t) with respect to x. The chain rule states:

d/dx ∫[a(x) to b(x)] f(t) dt = f(b(x)) * d(b(x))/dx - f(a(x)) * d(a(x))/dx

In this case, a(x) = 1 and b(x) = 3x.

Step 3: Now, differentiate the upper limit b(x) = 3x with respect to x:
d(b(x))/dx = d(3x)/dx = 3

Step 4: Differentiate the lower limit a(x) = 1 with respect to x:
d(a(x))/dx = d(1)/dx = 0 (since it's a constant)

Step 5: Finally, substitute these values into the Chain Rule formula:
dy/dx = f(3x) * 3 - f(1) * 0

Step 6: Simplify the expression to get the final derivative.

Now, to find the second derivative, d2y/dx2, you need to differentiate the first derivative, dy/dx, with respect to x using the chain rule again.

I hope this explanation helps you understand the process of finding the derivatives of the given function.