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Chapter 29: Problem 33

Evaluate each expression. $$f^{\prime \prime}(0) \text { where } f(t)=e^{\sin t} \cos t$$

Short Answer

Expert verified
After differentiating \(e^{\sin t} \cos t\) twice and evaluating at \(t = 0\), \(f''(0) = -1\).

Step by step solution

01

Differentiate the function once

Apply the product rule, which states that \(d(uv)/dt = u dv/dt + v du/dt\), to find the first derivative of \(f(t)=e^{\sin t} \cos t\).
02

Differentiate the function a second time

Take the derivative of the first derivative using the product rule again along with the chain rule for the exponential function.
03

Evaluate the second derivative at \(t = 0\)

Substitute \(t = 0\) into the second derivative to calculate \(f''(0)\).

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Key Concepts

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

Product Rule
Understanding the product rule is essential when dealing with the differentiation of functions that are the product of two or more functions. When you have a function expressed as the multiplication of two functions, for instance, an equation like f(t) = g(t) * h(t), the product rule allows you to differentiate it. According to the product rule, the derivative of this function is f'(t) = g'(t) * h(t) + g(t) * h'(t). This means that to find the derivative of the entire function, you must take the derivative of each function independently and add the two products.

Chain Rule
The chain rule comes into play when you are differentiating a composite function. In other words, when one function is nested inside another, such as f(g(t)). The chain rule states that the derivative of such a function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically, it鈥檚 expressed as (f(g(t)))' = f'(g(t)) * g'(t). This rule is particularly handy when dealing with exponential functions that have expressions more complicated than a simple variable for their exponents, such as e^{sin(t)} in the exercise.

Exponential Functions Differentiation
Exponential functions of the form f(t) = e^x, where e is the base of the natural logarithm, have a unique property: their derivative is the same as the original function. However, when the exponent is a function of t rather than just t, you must apply the chain rule. For example, if f(t) = e^{g(t)}, then f'(t) = g'(t) * e^{g(t)}. During differentiation of such functions, it's crucial to identify the inner function g(t) and use its derivative correctly as part of the process.

First Derivative Calculation
Calculating the first derivative of a function is the initial step in understanding the function's rate of change at any point. It involves applying differentiation rules like the product rule, the chain rule, or others specific to the function type like power or exponential rules. In the context of the given exercise, applying the product rule results in the first derivative of f(t) = e^{sin(t)} * cos(t). This includes taking the derivative of each component function, e^{sin(t)} and cos(t), and combining them according to the product rule. The first derivative is foundational for any further differentiation steps, such as finding the second derivative or the function's concavity.

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