Question
Use part one of the fundamental theorem of calculus to find the derivative of the function.
g(x) =
x
t4 + t6
dt
0
g'(x) =
g(x) =
x
t4 + t6
dt
0
g'(x) =
Answers
GPT-4o mini
To find the derivative of the function \( g(x) = \int_0^x (t^4 + t^6) \, dt \) using part one of the Fundamental Theorem of Calculus, we simply need to differentiate \( g(x) \) with respect to \( x \).
According to the Fundamental Theorem of Calculus, if \( G(t) \) is an antiderivative of \( f(t) \), then:
\[
\frac{d}{dx} \int_a^x f(t) \, dt = f(x)
\]
In this case, we have \( f(t) = t^4 + t^6 \). Therefore, we can compute the derivative of \( g(x) \):
\[
g'(x) = \frac{d}{dx} \int_0^x (t^4 + t^6) \, dt = x^4 + x^6
\]
So, the derivative is:
\[
g'(x) = x^4 + x^6
\]
According to the Fundamental Theorem of Calculus, if \( G(t) \) is an antiderivative of \( f(t) \), then:
\[
\frac{d}{dx} \int_a^x f(t) \, dt = f(x)
\]
In this case, we have \( f(t) = t^4 + t^6 \). Therefore, we can compute the derivative of \( g(x) \):
\[
g'(x) = \frac{d}{dx} \int_0^x (t^4 + t^6) \, dt = x^4 + x^6
\]
So, the derivative is:
\[
g'(x) = x^4 + x^6
\]