In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
10. y = ln(t^(3/2))

Verified step by step guidance

1. Recognize that the function is given as \(y = \ln\left(t^{\frac{3}{2}}\right)\), which is a natural logarithm of a power function.
2. Use the logarithmic property that allows you to bring the exponent in front: \(\ln\left(t^{\frac{3}{2}}\right) = \frac{3}{2} \ln(t)\).
3. Rewrite the function as \(y = \frac{3}{2} \ln(t)\) to simplify differentiation.
4. Recall the derivative of \(\ln(t)\) with respect to \(t\) is \(\frac{1}{t}\), so apply the constant multiple rule to get \(\frac{dy}{dt} = \frac{3}{2} \cdot \frac{1}{t}\).
5. Express the derivative as \(\frac{dy}{dt} = \frac{3}{2t}\), which is the derivative of the original function with respect to \(t\).

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation involves applying the derivative rules to functions involving logarithms. The derivative of ln(u), where u is a function of the variable, is (1/u) times the derivative of u. This technique simplifies differentiation of functions expressed as logarithms.

Power Rule for Exponents

The power rule states that the derivative of t^n with respect to t is n * t^(n-1). This rule is essential when differentiating expressions where the variable is raised to a constant exponent, such as t^(3/2) in the given function.

Properties of Logarithms

Logarithmic properties, like ln(a^b) = b * ln(a), allow simplification of expressions before differentiation. Applying these properties can transform complex logarithmic functions into simpler forms, making differentiation more straightforward.