 #1
 1,273
 80
Homework Statement:

If $$y=\left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^3$$
Find ##\tfrac{dy}{dx}##
Relevant Equations:

I hope it is ok for me to post here, I just needed ##\LaTeX##. Thank you!
FTC, chain rule, power rule
I hope it is ok for me to borrow this post: No HW help required here. I just wanted to have LaTeX enabled forum to post my answer to this thread in r/calculus
Work: If $$y= \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^3 $$
then $$\begin{gathered} \tfrac{dy}{dx} =3 \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^2\cdot \tfrac{d}{dx} \left( \int_0^x (t^{3}+1)^{10}\, dt\right) \quad (1)\\ =\boxed{3 \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^2\cdot (x^{3}+1)^{10}}\quad (2) \\ \end{gathered}$$
where (1) is power rule followed by the chain rule and (2) is FTC.
Work: If $$y= \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^3 $$
then $$\begin{gathered} \tfrac{dy}{dx} =3 \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^2\cdot \tfrac{d}{dx} \left( \int_0^x (t^{3}+1)^{10}\, dt\right) \quad (1)\\ =\boxed{3 \left( \int_0^x (t^{3}+1)^{10}\, dt\right) ^2\cdot (x^{3}+1)^{10}}\quad (2) \\ \end{gathered}$$
where (1) is power rule followed by the chain rule and (2) is FTC.