Use the Fundamental Theorem of Calculus to find G'(x) if:
/x²
G(x)= / cos t dt
/
1
change all the t's to x^2, then multiply by (x^2)'
G(x) = cos(x^2)(2x)
To find the derivative of `G(x)` using the Fundamental Theorem of Calculus, we will apply the Second Part of the Fundamental Theorem of Calculus, which states:
If `G(x) = ∫[a,x] f(t) dt`, where `a` is some constant, then `G'(x) = f(x)`.
In this case, we have `G(x) = ∫[1, x²] cos(t) dt`, where `a = 1` and `f(t) = cos(t)`.
To find `G'(x)`, we need to evaluate `f(x)`, where `x` is the upper limit of the integral. Therefore, we need to replace `t` with `x²` in `cos(t)`.
So, `G'(x) = cos(x²)`.