`y = int_0^(x^4)(cos^2 (theta))d theta` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

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Borys Shumyatskiy eNotes educator| Certified Educator

Hello!

Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f`
`F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.`

Here `f(t)=cos^2(t)` and `y(x)=F_0(x^4).`

 

Therefore

`y'(x)=d/(dx)(y(x))=d/(dx)(F_0(x^4))=F'_0(x^4)*4x^3=f(x^4)*4x^3=`

`=cos^2(x^4)*4x^3.`