Find g'(x)

G(x)=9tcos(t^10)dt; intergral from [0 to sqrt(x)]

If the entire function g(x) is an integral with upper and lower limits, that makes it a definite integral. A definite integral always has a constant value.

That would make g(x) a constant value. So what would its derivative be?

(This might be incorrect if my interpretation of the question is inaccurate)

using the 2nd FTC,

g'(x) = 9√x cos(x^5) * 1/(2√x)

If G(x) = ∫[a,u(x)] f(t) dt
G'(x) = f(u(x)) * u'(x)

This is just the chain rule, seen from the other side.

Note that

if F'(x) = f(x)
and G(x) = ∫[u(x),v(x)] f(t) dt
= F(v)-F(u)
so
G'(x) = f(v)*v' - f(u)*u'

If, as in your problem, the lower limit is a constant a, then F(a) is a constant, so F'(a) = 0 (that means that u is a constant, so u'=0 in the more general formula)

To find g'(x), we need to differentiate the integral of G(x) with respect to x. However, since the upper limit of integration in the given integral is a function of x, we will use the Fundamental Theorem of Calculus along with the Chain Rule.

Let's start by rewriting the integral using a different variable:

G(x) = ∫[0 to sqrt(x)] 9t*cos(t^10) dt

Now, let's differentiate G(x) with respect to x:

g'(x) = d/dx [ ∫[0 to sqrt(x)] 9t*cos(t^10) dt ]

According to the Fundamental Theorem of Calculus, if 'F(x)' is an antiderivative of 'f(x)', then:

d/dx [ ∫[a to b] f(x) dx ] = f(b) * db/dx - f(a) * da/dx

In our case, we have:

f(t) = 9t * cos(t^10)
a = 0
b = sqrt(x)

Now, let's evaluate the derivative using the Fundamental Theorem of Calculus and the Chain Rule:

g'(x) = f(sqrt(x)) * d(sqrt(x))/dx - f(0) * d(0)/dx

To find d(sqrt(x))/dx, we can differentiate sqrt(x) with respect to x:

d(sqrt(x))/dx = (1/2) * x^(-1/2)

Plugging in the values:

g'(x) = 9(sqrt(x)) * cos((sqrt(x))^10) * (1/2) * x^(-1/2) - 9(0) * (0)

Simplifying and removing the unnecessary terms:

g'(x) = (9/2) * sqrt(x) * cos(sqrt(x)^10) * x^(-1/2)

Therefore, the derivative of G(x) with respect to x, or g'(x), is:

g'(x) = (9/2) * sqrt(x) * cos(sqrt(x)^10) * x^(-1/2)