6.] Find (d)/(dt)int_(2)^(t^(4))e^(x^(2))dx

Apply Leibniz Rule: We are asked to find the derivative with respect to $ t $ of the integral of $ e^{x^2} $ from 2 to $ t^4 $. This is an application of the Leibniz rule, which states that if we have an integral of the form $ \int_{a(t)}^{b(t)} f(x, t) \, dx $, then its derivative with respect to $ t $ is given by $ \frac{d}{dt} \int_{a(t)}^{b(t)} f(x, t) \, dx = f(b(t), t) \cdot b'(t) - f(a(t), t) \cdot a'(t) $. In our case, $ f(x, t) = e^{x^2} $, $ a(t) = 2 $, and $ b(t) = t^4 $. Since $ a(t) $ is a constant, its derivative is 0, and we only need to consider the derivative of $ b(t) $ with respect to $ t $.Find Derivative of b(t): First, we find the derivative of $ b(t) = t^4 $ with respect to $ t $. Using the power rule, we get $ b'(t) = 4t^3 $.Apply Leibniz Rule: Now, we apply the Leibniz rule. Since $ a(t) = 2 $ is constant, its derivative is 0 and does not contribute to the derivative of the integral. Therefore, we have: $ \frac{d}{dt} \int_{2}^{t^4} e^{x^2} \, dx = e^{(t^4)^2} \cdot 4t^3 - 0 $.Simplify Expression: Simplify the expression to get the final answer. We have: $ \frac{d}{dt} \int_{2}^{t^4} e^{x^2} \, dx = e^{t^8} \cdot 4t^3 $.

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6.] Find
(d)/(dt)int_(2)^(t^(4))e^(x^(2))dx

Find $\frac{d}{dt}\int_{2}^{t^{4}}e^{x^{2}}dx$

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Evaluate definite integrals using the chain rule

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Q. Find

\frac{d}{dt}\int_{2}^{t^{4}}e^{x^{2}}dx

Apply Leibniz Rule: We are asked to find the derivative with respect to $t$ of the integral of $e^{x^2}$ from $2$ to $t^4$ . This is an application of the Leibniz rule, which states that if we have an integral of the form $\int_{a(t)}^{b(t)} f(x, t) \, dx$ , then its derivative with respect to $t$ is given by $\frac{d}{dt} \int_{a(t)}^{b(t)} f(x, t) \, dx = f(b(t), t) \cdot b'(t) - f(a(t), t) \cdot a'(t)$ . In our case, $f(x, t) = e^{x^2}$ , $a(t) = 2$ , and $b(t) = t^4$ . Since $a(t)$ is a constant, its derivative is $0$ , and we only need to consider the derivative of $e^{x^2}$ $0$ with respect to $t$ .
Find Derivative of b(t): First, we find the derivative of $b(t) = t^4$ with respect to $t$ . Using the power rule, we get $b'(t) = 4t^3$ .
Apply Leibniz Rule: Now, we apply the Leibniz rule. Since $a(t) = 2$ is constant, its derivative is $0$ and does not contribute to the derivative of the integral. Therefore, we have: $\newline$ $\frac{d}{dt} \int_{2}^{t^4} e^{x^2} \, dx = e^{(t^4)^2} \cdot 4t^3 - 0$ .
Simplify Expression: Simplify the expression to get the final answer. We have: $\newline$ $\frac{d}{dt} \int_{2}^{t^4} e^{x^2} \, dx = e^{t^8} \cdot 4t^3$ .