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For $t>0$, the derivative of the upper limit $u=\sqrt{t}$ with respect to $t$ is $1/(2\sqrt{t})$. This result is obtained by applying the power rule to $t^{1/2}$ and serves as the inner function's derivative factor in the chain rule when differentiating integrals with this variable upper limit.

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

For $t>0$, the derivative of the upper limit $u=\sqrt{t}$ with respect to $t$ is $1/(2\sqrt{t})$. This result is obtained by applying the power rule to $t^{1/2}$ and serves as the inner function's derivative factor in the chain rule when differentiating integrals with this variable upper limit.

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

The general formula for Euler's totient function is $\varphi(n) = n \prod_{i=1}^{m} \left(1 - \frac{1}{p_i}\right)$, where $p_1, p_2, \ldots, p_m$ are the distinct prime factors of $n$. This formula is derived by combining the multiplicativity of $\varphi$, the prime power formula $\varphi(p^k) = p^k - p^{k-1}$, and the Fundamental Theorem of Arithmetic.

Conditions: $n$ is a positive integer greater than 1.; $p_1, \ldots, p_m$ are all the distinct prime factors of $n$.

The general formula for Euler's totient function is $\varphi(n) = n \prod_{i=1}^{m} \left(1 - \frac{1}{p_i}\right)$, where $p_1, p_2, \ldots, p_m$ are the distinct prime factors of $n$. This formula is derived by combining the multiplicativity of $\varphi$, the prime power formula $\varphi(p^k) = p^k - p^{k-1}$, and the Fundamental Theorem of Arithmetic.

Conditions: $n$ is a positive integer greater than 1.; $p_1, \ldots, p_m$ are all the distinct prime factors of $n$.

To differentiate an integral whose upper limit is $u=\sqrt{t}$, apply the chain rule by evaluating the continuous integrand at $u$ and multiplying by the derivative of the upper limit, $1/(2\sqrt{t})$. This process requires $t > 0$ to ensure the upper limit function is differentiable.

Conditions: The integrand is continuous on the relevant interval.; The upper limit is a differentiable function of $t$, specifically $u=\sqrt{t}$.; $t > 0$.

To differentiate an integral whose upper limit is $u=\sqrt{t}$, apply the chain rule by evaluating the continuous integrand at $u$ and multiplying by the derivative of the upper limit, $1/(2\sqrt{t})$. This process requires $t > 0$ to ensure the upper limit function is differentiable.

Conditions: The integrand is continuous on the relevant interval.; The upper limit is a differentiable function of $t$, specifically $u=\sqrt{t}$.; $t > 0$.