What would you like to understand?

The derivative of the upper limit $u=\sqrt{t}$ with respect to $t$ is $1/(2\sqrt{t})$. This factor is used in the chain rule when differentiating an integral with this variable upper limit.

Conditions: $t > 0$.; The upper limit is defined as $u=\sqrt{t}$.

The derivative of the upper limit $u=\sqrt{t}$ with respect to $t$ is $1/(2\sqrt{t})$. This factor is used in the chain rule when differentiating an integral with this variable upper limit.

Conditions: $t > 0$.; The upper limit is defined as $u=\sqrt{t}$.

The simplified derivative is $\sin(t^{5/2})/(2t^{5/2})$. This is obtained by evaluating the integrand at $x=\sqrt{t}$ and multiplying by the derivative of the upper limit, $1/(2\sqrt{t})$.

Conditions: The integrand is $\sin(x^5)/x^4$.; The lower limit is the constant 2.; The upper limit is $\sqrt{t}$.; $t > 0$.

The simplified derivative is $\sin(t^{5/2})/(2t^{5/2})$. This is obtained by evaluating the integrand at $x=\sqrt{t}$ and multiplying by the derivative of the upper limit, $1/(2\sqrt{t})$.

Conditions: The integrand is $\sin(x^5)/x^4$.; The lower limit is the constant 2.; The upper limit is $\sqrt{t}$.; $t > 0$.

The net-change formula states that for a continuously differentiable function on an interval, integrating its derivative gives the endpoint difference. Specifically, the integral of the derivative $f'$ from $a$ to $b$ equals $f(b) - f(a)$.

Conditions: The function is continuously differentiable on the interval.; The integration is over the interval from $a$ to $b$.

The net-change formula states that for a continuously differentiable function on an interval, integrating its derivative gives the endpoint difference. Specifically, the integral of the derivative $f'$ from $a$ to $b$ equals $f(b) - f(a)$.

Conditions: The function is continuously differentiable on the interval.; The integration is over the interval from $a$ to $b$.