Problem 46
Question
In each of Exercises \(45-50\) use the Chain Rule repeatedly to determine the derivative with respect to \(x\) of the given expression. $$ \sin (\cos (3 x)) $$
Step-by-Step Solution
Verified Answer
The derivative is \(-3 \sin(3x) \cos(\cos(3x))\).
1Step 1: Identify the Composition of Functions
First, recognize that the given expression \( \sin(\cos(3x)) \) is a composition of three functions: an outer function \( \sin(u) \), an intermediate function \( \cos(v) \), and an innermost function \( 3x \). Here \( u = \cos(3x) \) and \( v = 3x \).
2Step 2: Differentiate the Outer Function
Apply the Chain Rule starting from the outermost function. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \). Thus, the first part of the derivative is \( \cos(\cos(3x)) \cdot \frac{du}{dx} \).
3Step 3: Differentiate the Intermediate Function
Now differentiate the intermediate function \( \cos(v) \) with respect to \( v \). The derivative is \(-\sin(v) \), giving \( \frac{du}{dv} = -\sin(3x) \). Thus, the expression becomes \( \cos(\cos(3x)) \cdot (-\sin(3x)) \cdot \frac{dv}{dx} \).
4Step 4: Differentiate the Innermost Function
Finally, differentiate the innermost function \( 3x \) with respect to \( x \), which gives us \( \frac{dv}{dx} = 3 \). Incorporate this into the expression, resulting in \( \cos(\cos(3x)) \cdot (-\sin(3x)) \cdot 3 \).
5Step 5: Final Calculation
Combine all parts: the derivative of \( \sin(\cos(3x)) \) with respect to \( x \) is \( -3 \cos(\cos(3x)) \sin(3x) \). Simplifying, we have: \[ -3 \sin(3x) \cos(\cos(3x)). \]
Key Concepts
DerivativeComposition of FunctionsDifferentiation
Derivative
The concept of a derivative is foundational in calculus, and it represents the rate of change of a function concerning its variable.
The function's derivative defines how the function's value changes as its input changes, essentially capturing the function's behavior.
The notation for a derivative can vary, including formats like \( f'(x) \), \( \frac{dy}{dx} \), or \( \frac{df(x)}{dx} \).
Some key points about derivatives include:
The function's derivative defines how the function's value changes as its input changes, essentially capturing the function's behavior.
The notation for a derivative can vary, including formats like \( f'(x) \), \( \frac{dy}{dx} \), or \( \frac{df(x)}{dx} \).
Some key points about derivatives include:
- A derivative at a specific point can be viewed as the slope of the tangent line to the curve at that point.
- Derivatives are crucial in many fields, like physics for motion analyses or in economics for modeling cost or profit changes.
- Calculating derivatives is a skill that unlocks advanced mathematical applications and provides deeper insight into function characteristics.
Composition of Functions
The composition of functions involves combining two or more functions, making complex expressions wherein the output of one becomes the input of another.
This concept is essential for understanding how different processes or phenomena relate to one another.
In the expression \( \sin(\cos(3x)) \), we see a classic example of function composition.
Here, the innermost function is \( 3x \), the next is \( \cos(3x) \), and finally, the outermost function is \( \sin(u) \).
When dealing with compositions:
This concept is essential for understanding how different processes or phenomena relate to one another.
In the expression \( \sin(\cos(3x)) \), we see a classic example of function composition.
Here, the innermost function is \( 3x \), the next is \( \cos(3x) \), and finally, the outermost function is \( \sin(u) \).
When dealing with compositions:
- Identify the order of functions from innermost to outermost, as this enables systematic differentiation and analysis.
- Understand that differentiating compositions often requires clever techniques like the Chain Rule due to their interdependent nature.
- Function compositions are prevalent in both practical applications and theoretical settings, helping relate various model behaviors and results.
Differentiation
Differentiation refers to the process of finding a derivative, essential for grasping the dynamic behavior of functions.
It involves applying various rules and methodologies to calculate the derivative accurately.
The Chain Rule is a critical technique in differentiation, especially pertinent when dealing with compositions of functions.
Key points about differentiation include:
It involves applying various rules and methodologies to calculate the derivative accurately.
The Chain Rule is a critical technique in differentiation, especially pertinent when dealing with compositions of functions.
Key points about differentiation include:
- Basic rules like the product and quotient rules provide starting points, but advanced functions often require more sophisticated approaches like the Chain Rule or Implicit Differentiation.
- Differentiation allows us to calculate velocities, accelerations, or any rate of change, making it vital for applications in science and engineering.
- When using the Chain Rule, differentiate each function layer step by step, then apply the derivatives' product to find the overall derivative.
Other exercises in this chapter
Problem 46
Find the tangent line to the parametric curve \(x=\varphi_{1}(t), y=\varphi_{2}(t)\) at the point corresponding to the given value \(t_{0}\) of the parameter. $
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Find a function whose derivative is the given function. \(3 / x^{3}\)
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Find the tangent line to the graph of the given function at the given point. $$ f(x)=x \cos (x), \quad P=(\pi,-\pi) $$
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