In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
18. y = t√(ln t)

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Identify the function given: \(y = t \sqrt{\ln t}\). Notice that this is a product of two functions of \(t\): \(t\) and \(\sqrt{\ln t}\).

Rewrite the square root in exponent form to make differentiation easier: \(y = t (\ln t)^{1/2}\).

Apply the product rule for differentiation, which states: \(\frac{d}{dt}[u v] = u' v + u v'\), where \(u = t\) and \(v = (\ln t)^{1/2}\).

Find the derivatives of each part: \(u' = \frac{d}{dt}[t] = 1\), and for \(v'\), use the chain rule. First, write \(v = (\ln t)^{1/2}\), so \(v' = \frac{1}{2} (\ln t)^{-1/2} \cdot \frac{d}{dt}[\ln t]\). Since \(\frac{d}{dt}[\ln t] = \frac{1}{t}\), combine these to get \(v' = \frac{1}{2} (\ln t)^{-1/2} \cdot \frac{1}{t}\).

Substitute \(u\), \(u'\), \(v\), and \(v'\) back into the product rule formula: \(\frac{dy}{dt} = 1 \cdot (\ln t)^{1/2} + t \cdot \left( \frac{1}{2} (\ln t)^{-1/2} \cdot \frac{1}{t} \right)\). Simplify the expression where possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of a Product

When a function is the product of two functions, such as y = t * √(ln t), the derivative is found using the product rule. This rule states that the derivative of f(t)g(t) is f'(t)g(t) + f(t)g'(t), requiring differentiation of each part separately.

Chain Rule

The chain rule is used to differentiate composite functions, like √(ln t). It involves taking the derivative of the outer function evaluated at the inner function and multiplying by the derivative of the inner function. For example, d/dt [√(ln t)] requires differentiating the square root and the natural log inside.

Derivative of Logarithmic Functions

The derivative of the natural logarithm function ln t is 1/t. This is essential when differentiating expressions involving ln t, especially when combined with other functions, as in √(ln t). Recognizing this derivative helps simplify the application of the chain rule.

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