Problem 88 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=t \log _{3}\left(e^{(\sin t)(\ln 3)}\right)$$

Step-by-Step Solution

Answer

The derivative is \( t \cos t + \sin t \).

Step 1: Identify the Components

The given function is \( y = t \log_{3}\left(e^{(\sin t)(\ln 3)}\right) \). This is a product of \( t \) and a logarithmic function \( \log_3(\cdot) \). The natural base is transformed using an exponentiation that involves a trigonometric function \( \sin t \).

Step 2: Simplify the Logarithmic Expression

The expression inside the logarithm can be simplified. Since \( e^{(\sin t)(\ln 3)} = (e^{\ln 3})^{\sin t} = 3^{\sin t} \), the function becomes \( y = t \log_{3}(3^{\sin t}) \).

Step 3: Apply Logarithm Property

Use the property \( \log_b(a^c) = c \log_b(a) \) to simplify: \( \log_3(3^{\sin t}) = \sin t \cdot \log_3(3) = \sin t \). Therefore, \( y = t \sin t \).

Step 4: Differentiate Using Product Rule

The function is now \( y = t \sin t \). Use the product rule for derivatives, \( \frac{d}{dt}[uv] = u'v + uv' \), where \( u = t \) and \( v = \sin t \). Then, \( u' = 1 \) and \( v' = \cos t \).

Step 5: Compute the Derivative

Applying the product rule: \( \frac{d}{dt}(t \sin t) = t(\cos t) + (1)(\sin t) = t \cos t + \sin t \).

Step 6: Solution and Simplify if Needed

The derivative of \( y = t \log_3(e^{(\sin t)(\ln 3)}) \) with respect to \( t \) is \( \frac{dy}{dt} = t \cos t + \sin t \).

Key Concepts

Product Rule

When you are trying to find the derivative of a product of two functions, the **product rule** is your go-to tool. It works seamlessly whenever you have two functions, say \( u \) and \( v \), multiplied together, and you need to differentiate them. The rule tells us:

• \( \frac{d}{dt}[uv] = u'v + uv' \)

This means you differentiate the first function, \( u \), while keeping the second function, \( v \), intact; then you keep \( u \) intact and differentiate the second function. Finally, you sum these two products. This rule is especially handy when working with problems involving products of polynomials, exponentials, and trigonometric functions.

In the given exercise, using the product rule was necessary because we had the function \( y = t \sin t \). By identifying the parts \( u = t \) and \( v = \sin t \), the solution was able to calculate the derivative correctly by applying the product rule:

• \( t \cdot \cos t + 1 \cdot \sin t \)

Getting comfortable with this rule, ensures you handle the derivation of products confidently.

Logarithm Properties

Logarithms are a powerful mathematical tool that can greatly simplify functions, especially when dealing with exponents. One essential property to remember is how to handle exponents inside a logarithm, using the formula:

• \( \log_b(a^c) = c \cdot \log_b(a) \)

This property tells us that when you have an exponent inside a logarithm, you can bring the exponent down in front as a multiplier. It helps break down complex expressions into more manageable pieces.

In our exercise, the expression \( \log_3(3^{\sin t}) = \sin t \cdot \log_3(3) \) simplifies further because \( \log_3(3) = 1 \), resulting simply in \( \sin t \). This simplification was crucial in moving to the next step of differentiating the function. Understanding these properties makes it easier to progress through logarithmic and exponential derivations efficiently.

Trigonometric Functions

Among the many functions in calculus, trigonometric functions like \( \sin t \) and \( \cos t \) play a crucial role. They appear frequently in problems involving waves, oscillations, and rotational movements.

When differentiating trigonometric functions, knowing the derivatives is key:

• The derivative of \( \sin t \) is \( \cos t \)
• The derivative of \( \cos t \) is \(-\sin t \)

This knowledge allows you to tackle more complex derivative problems. In the exercise, after simplifying the function to \( y = t \sin t \), the differentiation of \( \sin t \) was straightforward, playing a significant role in finding the final derivative. As you work with these functions, practicing their derivatives helps in mastering how they interact with other elements of calculus, such as products and logarithms.