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- 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:

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.

**No HW help required here. I just wanted to have LaTeX enabled forum to post my answer to**this thread in r/calculusWork: 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.