7–28. Derivatives Evaluate the following derivatives.


d/dt ((sin t)^{√t})

Verified step by step guidance

1. Step 1: Recognize that the given function is of the form \( f(t) = (\sin t)^{\sqrt{t}} \), which is a composite function involving both a power and a trigonometric function. To differentiate it, we will use the logarithmic differentiation technique.
2. Step 2: Take the natural logarithm of both sides to simplify the differentiation process. Let \( y = (\sin t)^{\sqrt{t}} \), then \( \ln y = \sqrt{t} \cdot \ln(\sin t) \).
3. Step 3: Differentiate both sides with respect to \( t \). Use the chain rule on the left-hand side and the product rule on the right-hand side. The derivative of \( \ln y \) is \( \frac{1}{y} \cdot \frac{dy}{dt} \), and the derivative of \( \sqrt{t} \cdot \ln(\sin t) \) requires applying the product rule.
4. Step 4: Apply the product rule to \( \sqrt{t} \cdot \ln(\sin t) \). The derivative of \( \sqrt{t} \) is \( \frac{1}{2\sqrt{t}} \), and the derivative of \( \ln(\sin t) \) is \( \frac{1}{\sin t} \cdot \cos t \) (using the chain rule). Combine these results.
5. Step 5: Solve for \( \frac{dy}{dt} \) by multiplying through by \( y \), which is \( (\sin t)^{\sqrt{t}} \). Substitute \( y \) back into the expression to obtain the derivative in terms of \( t \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental differentiation technique used to find the derivative of composite functions. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for differentiating functions where one function is nested within another.

Product Rule

The Product Rule is a method for differentiating products of two functions. It states that if you have two functions u(t) and v(t), the derivative of their product is given by u'v + uv'. This rule is particularly useful when dealing with expressions where two functions are multiplied together, such as in the case of (sin t)^{√t}.

Exponential Functions and Logarithmic Differentiation

Exponential functions, especially those involving variables in the exponent, can be complex to differentiate directly. Logarithmic differentiation is a technique that simplifies this process by taking the natural logarithm of both sides of the equation. This allows for easier manipulation and differentiation, particularly when dealing with functions like (sin t)^{√t}, where the exponent is itself a function of t.