Problem 23 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. \begin{equation}y=e^{(\cos t+\ln t)}\end{equation}

Step-by-Step Solution

Answer

\(\frac{dy}{dt} = e^{(\cos t + \ln t)} \left(-\sin t + \frac{1}{t}\right)\)

Step 1: Identify the Function

We have the function \(y = e^{(\cos t + \ln t)}\). This is an exponential function where the exponent is a sum of \(\cos t\) and \(\ln t\). Our goal is to find \(\frac{dy}{dt}\).

Step 2: Apply the Chain Rule

To find the derivative of an exponential function \(e^u\), where \(u\) is a function of \(t\), we use the chain rule. If \(y = e^u\), then \(\frac{dy}{dt} = e^u \cdot \frac{du}{dt}\). Here, \(u = \cos t + \ln t\).

Step 3: Differentiate the Exponent

Differentiate \(u = \cos t + \ln t\) with respect to \(t\). The derivative of \(\cos t\) is \(-\sin t\), and the derivative of \(\ln t\) is \(\frac{1}{t}\). Thus, \(\frac{du}{dt} = -\sin t + \frac{1}{t}\).

Step 4: Compute the Derivative of y with Respect to t

Apply the chain rule: \(\frac{dy}{dt} = e^{(\cos t + \ln t)} \cdot \left(-\sin t + \frac{1}{t}\right)\). This is the expression for the derivative of \(y\) with respect to \(t\).

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus. It is used to differentiate composite functions, where one function is nested inside another. Consider a scenario where you have a function defined as \( y = f(g(t)) \). Here, \( g(t) \) is inside the function \( f \). The chain rule allows us to find the derivative of such a function by multiplying the derivative of the outer function by the derivative of the inner function.

In our exercise, we have the function \( y = e^{u} \), where \( u = \cos t + \ln t \). By applying the chain rule, we differentiate \( e^u \) with respect to \( t \). This becomes:

• First, differentiate \( e^u \) to get \( e^u \).
• Then, multiply by \( \frac{du}{dt} \), the derivative of \( u \).

So, \( \frac{dy}{dt} = e^{u} \cdot \frac{du}{dt} \). This helps in understanding the sequence in which to apply the chain rule effectively.

Exponential Functions

Exponential functions are an essential class of functions in mathematics. They are defined by the expression \( e^x \) or in general terms, \( a^x \), where \( a \) is a constant. The common exponential function is \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828.

These functions have unique properties, such as the derivative of \( e^x \) being \( e^x \) itself. This makes them mathematically interesting and useful in various real-life applications. In calculus, they often appear in growth and decay problems.

In the given exercise, the function \( y = e^{(\cos t + \ln t)} \) is an exponential function where the exponent is a combination of other functions. When tasked with differentiating this function, we apply the chain rule to find its derivative easily.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to simplify the process of differentiating certain complex functions. It involves taking the natural logarithm of both sides of an equation \( y = f(x) \), and then using properties of logarithms to simplify before differentiating.

This method is particularly useful when dealing with products, quotients, or powers of functions. By applying the logarithm, products become sums, quotients become differences, and powers can be brought down in front of the logarithm.

In our exercise, although we do not explicitly use logarithmic differentiation to find the derivative of \( y = e^{(\cos t + \ln t)} \), understanding how \( \ln t \) functions is crucial. The derivative of \( \ln t \) gives \( \frac{1}{t} \), which is a component of our chain rule application. This highlights the power of logs in transforming and simplifying differentiation tasks.