Problem 21 Find the derivative of $y$ wit... [FREE SOLUTION]

Chapter 7: Problem 21

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

Short Answer

Expert verified

The derivative is $ y' = e^{(\cos t + \ln t)} \left(-\sin t + \frac{1}{t}\right) $.

Step by step solution

Identify the Function Components

The given function is $ y = e^{(\cos t + \ln t)} $. The exponent consists of two separate functions: $ \cos t $ and $ \ln t $. We will apply the chain rule to find the derivative.

Differentiate Using the Chain Rule

To differentiate $ y = e^{u} $, where $ u = \cos t + \ln t $, we use the chain rule. The derivative of $ e^u $ with respect to $ t $ is $ e^u \cdot \frac{du}{dt} $.

Differentiate the Inner Function

Find the derivative of the inner function $ u = \cos t + \ln 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} $.

Apply the Chain Rule

Substitute $ \frac{du}{dt} = -\sin t + \frac{1}{t} $ back into the chain rule derivative $ y' = e^{(\cos t + \ln t)} \cdot \left(-\sin t + \frac{1}{t}\right) $.

Simplify the Expression

The expression can be simplified to $ y' = e^{(\cos t + \ln t)} \cdot \left(-\sin t + \frac{1}{t}\right) $. Thus, this is the derivative of the function with respect to $ t $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Chain Rule

The chain rule is a fundamental tool in calculus that helps us find the derivative of a composite function. A composite function is essentially a function inside another function, like our example with the exponential function raised to the power of a sum.

The outer function is the exponential function, represented as $ e^u $.
The inner function is $ u = \cos t + \ln t $. Here, $ u $ is a function of $ t $.

Using the chain rule, we differentiate the outer function $ e^u $ with respect to $ u $, which gives us $ e^u $.
Next, we multiply by the derivative of the inner function $ u $ with respect to $ t $, denoted as $ \frac{du}{dt} $. This gives us the complete derivative: $ \frac{dy}{dt} = e^u \cdot \frac{du}{dt} $.
Understanding each step ensures you can tackle more complicated derivative problems.

Exploring Exponential Functions

Exponential functions involve the mathematical constant $ e $, approximately equal to 2.71828. The function $ e^x $ is unique because its derivative is itself, meaning $ \frac{d}{dx}(e^x) = e^x $. This property makes it highly valuable in calculus.Moving to our exercise, where the exponential function is $ e^{(\cos t + \ln t)} $:

The base $ e $ plays a crucial role in maintaining the properties of the function, regardless of the complexity of the exponent $(\cos t + \ln t)$.
When differentiating, we keep the structure $ e^u $ intact while focusing on the exponent's derivative. This maintains the elegance of the exponential property.

In developmental terms, once we're comfortable with the derivative of $ e^x $, handling more complex scenarios becomes more approachable.

Working with Trigonometric Functions

Trigonometric functions, like $ \cos t $ and $ \sin t $, are common in calculus exercises because they model cyclical behavior. Knowing their derivatives is essential:

The derivative of $ \cos t $ is $ -\sin t $.
The derivative of $ \sin t $ is $ \cos t $.

In the given exercise, the term $ \cos t $ appears as a part of the composite function. To differentiate it, we find $ \frac{d}{dt}(\cos t) = -\sin t $.
Communicating effectively between trigonometric and exponential functions is key to simplifying derivative calculations.
Pairing these concepts with the chain rule allows us to navigate even more intricate calculus problems easily.

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91影视

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

Short Answer

Step by step solution

Identify the Function Components

Differentiate Using the Chain Rule

Differentiate the Inner Function

Apply the Chain Rule

Simplify the Expression

Key Concepts

Understanding the Chain Rule

Exploring Exponential Functions

Working with Trigonometric Functions

One App. One Place for Learning.

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