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Magazine Chapter 3: Problem 32
Find the derivatives of the functions. Assume \(a, b,\) and \(c\) are constants.$$w=e^{-\sin \theta}$$
Short Answer
Expert verified
\( \frac{dw}{d\theta} = -e^{- heta} \cos \theta \)
Step by step solution
01
Identify the Outer Function
The function given is \( w = e^{- heta} \). The outer function is an exponential function \( e^u \) where \( u = -\sin \theta \). The first step in differentiation is to apply the chain rule.
02
Differentiate the Outer Function
Differentiate the outer function \( e^u \) with respect to \( u \). The derivative of \( e^u \) is itself: \( \frac{d}{du}[e^u] = e^u \).
03
Identify the Inner Function
The inner function in this case is \( u = -\sin \theta \). We will now differentiate this with respect to \( \theta \).
04
Differentiate the Inner Function
Differentiate the inner function \( -\sin \theta \) with respect to \( \theta \). The derivative is \( -\cos \theta \), because \( \frac{d}{d\theta}[-\sin \theta] = -\cos \theta \).
05
Apply the Chain Rule
Now apply the chain rule, which states \( \frac{dw}{d\theta} = \frac{dw}{du} \cdot \frac{du}{d\theta} \). Substituting what we found, \( \frac{dw}{d\theta} = e^{- heta} \cdot (-\cos \theta) \).
06
Simplify the Result
Simplify the expression to get the final derivative. The derivative \( \frac{dw}{d\theta} = -e^{- heta} \cos \theta \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
The chain rule is a fundamental concept in calculus used for finding the derivative of a composite function. If you have a function that is made up of two or more nested functions, the chain rule allows us to differentiate it efficiently.
Imagine a function like our given case, where \( w = e^{-\sin \theta} \). It combines an exponential function and a trigonometric function. To differentiate it, we use the chain rule, which says:
Imagine a function like our given case, where \( w = e^{-\sin \theta} \). It combines an exponential function and a trigonometric function. To differentiate it, we use the chain rule, which says:
- Differentiate the outer function concerning its operand.
- Multiply by the derivative of the inner function.
Exponential Function
Exponential functions are functions that involve the constant \( e \), approximately equal to 2.718. These functions grow or decay at a rate proportional to their value, making them critical in modeling various natural processes.
In the context of derivatives, the exponential function is very special. If your function is \( e^u \), then its derivative with respect to \( u \) is simply \( e^u \). This property makes exponential functions very straightforward to differentiate, as seen in our solution where \( e^{-\sin \theta} \) retains its form when differentiated and just needs multiplication with the derivative of the inside term.
The simplicity of differentiating exponential functions is due to the unique property of \( e \), making it the natural base of exponential functions.
In the context of derivatives, the exponential function is very special. If your function is \( e^u \), then its derivative with respect to \( u \) is simply \( e^u \). This property makes exponential functions very straightforward to differentiate, as seen in our solution where \( e^{-\sin \theta} \) retains its form when differentiated and just needs multiplication with the derivative of the inside term.
The simplicity of differentiating exponential functions is due to the unique property of \( e \), making it the natural base of exponential functions.
Differentiation
Differentiation is the process in calculus of finding the derivative of a function, which represents the rate of change of the function with respect to one of its variables.
In our exercise, differentiation allows us to find out how \( w = e^{- sin \theta} \) changes as \( \theta \) changes. This involves:
In our exercise, differentiation allows us to find out how \( w = e^{- sin \theta} \) changes as \( \theta \) changes. This involves:
- Identifying both the inner and outer functions.
- Applying the chain rule to find the derivative.
Trigonometric Functions
Trigonometric functions like sine and cosine are basic functions in mathematics used to relate angles to side lengths in right-angled triangles. They also play a significant role in periodic phenomena, like waves.
In differentiation, trigonometric functions have distinct derivatives:
In differentiation, trigonometric functions have distinct derivatives:
- The derivative of \( \sin \theta \) is \( \cos \theta \).
- The derivative of \( \cos \theta \) is \(-\sin \theta \).
