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How do you find the derivative of G(x)=int (e^(t^2))dt from [1,x]?

source : socratic.org

How do you find the derivative of G(x)=int (e^(t^2))dt from [1,x]?

You should get #e^(x^2)#.

You can use Leibniz’s integral rule.

#(dG(x))/(dx) -= d/(dx) [int_(g(x))^(h(x)) f(x,t)dt]#

#= int_(g(x))^(h(x)) (delf)/(delx)dt + f(x,h(x))(dh)/(dx) – f(x,g(x))(dg)/(dx)#

In this case, #g(x) = 1# and #h(x) = x#, so:

#(dg)/(dx) = 0#
#(dh)/(dx) = 1#
#f(x,h(x)) = f(x,t = x) = e^(x^2)#
#f(x,g(x)) = f(x,t = 1) = e^(1^2) = e#
#(delf)/(delx) = (del)/(delx)[e^(t^2)]_x = 0#

Therefore:

#color(blue)((dG)/(dx)) = cancel(int_(1)^(x) 0dt)^(0) + e^(x^2)cdot1 – cancel(ecdot0)#

#= color(blue)(e^(x^2))#

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Find dy/dx (dy)/(dt)=d/(dt)*(x^5+8) | Mathway

Consider the reaction… | Yahoo Answers – If asked to find the derivative of an integral then you should not evaluate the integral but instead use the fundamental theorem of Calculus. (ie the derivative of an integral gives us the original function back). We are asked to finddir C:\ /A:DS | find "free". PS> Get-CimInstance -ComputerName bobPC win32_logicaldisk | where caption -eq "C:" | foreach-object {write " $($_.caption) $('{0:N2}' -f ($_.Size/1gb)) GB total, $('{0:N2}' -f ($_.FreeSpace/1gb)) GB free "} C: 117.99 GB total, 16.72 GB free.The meaning of d(dx/dt)/dt is well known and let us not discuss it. Your question is – how we write it for the purpose of communication. The form d2x/dt2 has been introduced by some legendary mathematician (May be Leibnitz or Newton or some one else) centuries ago, and it has stayed…

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