In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
37. ∫(from x²/2 to x²)ln(√t)dt

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Identify that the function is defined as an integral with variable limits: \(y = \int_{\frac{x^{2}}{2}}^{x^{2}} \ln(\sqrt{t}) \, dt\).

Recall the Leibniz rule for differentiation of an integral with variable limits: if \(y = \int_{a(x)}^{b(x)} f(t) \, dt\), then \(\frac{dy}{dx} = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\).

Determine the upper limit function \(b(x) = x^{2}\) and its derivative \(b'(x) = 2x\).

Determine the lower limit function \(a(x) = \frac{x^{2}}{2}\) and its derivative \(a'(x) = x\).

Evaluate the integrand at the limits: \(f(t) = \ln(\sqrt{t}) = \frac{1}{2} \ln(t)\), so compute \(f(b(x))\) and \(f(a(x))\), then apply the formula: \(\frac{dy}{dx} = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\).

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Fundamental Theorem of Calculus Part 1

This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable limit, its derivative is the integrand evaluated at that limit times the derivative of the limit. It allows us to differentiate integrals with variable limits directly.

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Fundamental Theorem of Calculus Part 1

Leibniz Rule for Differentiation Under the Integral Sign

Leibniz Rule generalizes the Fundamental Theorem by handling integrals with both upper and lower limits as functions of the variable. The derivative is the integrand evaluated at the upper limit times the derivative of the upper limit minus the integrand at the lower limit times the derivative of the lower limit.

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Additional Rules for Indefinite Integrals

Chain Rule

The chain rule is used to differentiate composite functions. When the limits of integration are functions of x, their derivatives require applying the chain rule to correctly compute the derivative of the integral with respect to x.

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