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.