Find F'(x) given F(x)= integral(upper 3x)(lower-3x)s^2 ds
Answer choices
A. F'(x)=2187x^6
B. F'(x)=54x^2
C. F'(x)=24x^2
D. F'(x)=729x^6
that would be (3x)^2 * (3) - (-3x)^2 * (-3)
so does that mean it would be A,B,C,or D????
@oobleck how did you get that?
Recall that if F(x) = ∫[u,v] f(s) ds
then
F'(x) = f(v) * v' - f(u) * u'
This is just the Chain Rule in reverse
To find the derivative of a function, we can use the Fundamental Theorem of Calculus. According to this theorem, if a function F(x) is defined as the integral of another function f(x), then the derivative of F(x) can be found by evaluating f(x) at x and multiplying by the derivative of the upper bound of integration.
In this case, F(x) is defined as the integral of s^2 with respect to s, where the upper bound of integration is 3x and the lower bound is -3x. So, we first need to find the antiderivative of s^2.
The antiderivative of s^2 is (1/3)s^3. Therefore, we have:
F(x) = integral from -3x to 3x of s^2 ds = (1/3)(3x)^3 - (1/3)(-3x)^3 = (1/3)(27x^3) - (1/3)(-27x^3) = (27/3)x^3 + (27/3)x^3 = 18x^3.
Now, we can find the derivative of F(x) by applying the chain rule. The chain rule states that if we have a composite function, we need to multiply the derivative of the outer function by the derivative of the inner function. In this case, the outer function is 18x^3 and the inner function is 3x.
Using the chain rule, we get:
F'(x) = 18x^3 * d/dx(3x) = 18x^3 * 3 = 54x^3.
Therefore, the correct answer choice is B. F'(x) = 54x^2.