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Magazine Chapter 3: Problem 45
Find \(d y / d t\) $$y=(t \tan t)^{10}$$
Short Answer
Expert verified
\( \frac{dy}{dt} = 10 (t \tan t)^9 (\tan t + t \sec^2 t) \).
Step by step solution
01
Recognize the Chain Rule
The given function is \(y = (t \tan t)^{10}\). This is a composite function, where the outer function is \(u^{10}\) with \(u = t \tan t\), and the inner function is the \(t \tan t\). The chain rule can be applied here. The derivative of \(u^{10}\) with respect to \(t\) is \(10u^9 \frac{du}{dt}\).
02
Differentiate Outer Function
Differentiate the outer function \(u^{10}\) with respect to \(u\). This gives \(10u^9\).
03
Differentiate Inner Function
Differentiate the inner function \(u = t \tan t\). Use the product rule here since \(u\) is a product of two functions, \(t\) and \(\tan t\). So, \(\frac{du}{dt} = \frac{d}{dt}(t) \cdot \tan t + t \cdot \frac{d}{dt}(\tan t)\).
04
Apply the Product Rule
Compute \(\frac{du}{dt}\):- The derivative of \(t\) is \(1\).- The derivative of \(\tan t\) is \(\sec^2 t\).So using the product rule: \(\frac{du}{dt} = 1 \cdot \tan t + t \cdot \sec^2 t = \tan t + t \sec^2 t\).
05
Combine Results Using the Chain Rule
Now plug the derivatives back into the chain rule result:\[ \frac{dy}{dt} = 10 (t \tan t)^9 \cdot (\tan t + t \sec^2 t) \]This combines the result of differentiating the outer and inner functions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Composite Function
In calculus, a composite function is essentially a function within a function. In the expression given, \( y = (t \tan t)^{10} \), the composite structure becomes clear. Here, the inner function is \( u = t \tan t \), while the outer function is \( u^{10} \). The role of composite functions is crucial in understanding and applying the chain rule effectively.
Recognizing composite functions helps streamline the differentiation process. It allows us to break down a complex function into simpler parts that are more manageable. When working with composite functions, you address the outer function and inner function separately before combining results.
Recognizing composite functions helps streamline the differentiation process. It allows us to break down a complex function into simpler parts that are more manageable. When working with composite functions, you address the outer function and inner function separately before combining results.
Product Rule
When differentiating a function that is a product of two other functions, the product rule is your go-to tool. For the given exercise, we need it to differentiate \( t \tan t \), which is a product of \( t \) and \( \tan t \).
The product rule states: If you have \( u(t) = v(t) \, w(t) \), then \( \frac{du}{dt} = v'(t) w(t) + v(t) w'(t) \).
For \( u = t \tan t \):
The product rule states: If you have \( u(t) = v(t) \, w(t) \), then \( \frac{du}{dt} = v'(t) w(t) + v(t) w'(t) \).
For \( u = t \tan t \):
- The derivative of \( t \) is \( 1 \).
- The derivative of \( \tan t \) is \( \sec^2 t \).
Differentiation
Differentiation is at the heart of the calculus problem presented. It allows us to compute the rate at which one variable changes concerning another. For \( y = (t \tan t)^{10} \), the differentiation involves applying both the chain and the product rules.
Understanding differentiation involves recognizing different types of functions and the rules applied to them. Whether it's dealing with composite functions or products of functions, having a firm grasp of differentiation rules like the chain rule and product rule provides the tools to solve problems effectively and efficiently.
Understanding differentiation involves recognizing different types of functions and the rules applied to them. Whether it's dealing with composite functions or products of functions, having a firm grasp of differentiation rules like the chain rule and product rule provides the tools to solve problems effectively and efficiently.
Calculus Problem Solving
Effective calculus problem solving often involves a series of straightforward steps:
Recognize and differentiate each layer of the function until you arrive at the complete derivative, ensuring each piece is precisely calculated and combined in your final answer.
- Identify the types of functions involved: Here, we have a composite function.
- Choose the correct rule for differentiation: Apply the chain rule for composite functions and the product rule for products of functions.
- Perform each step systematically: Differentiate the outer function, then the inner function, and finally combine results using the chain rule.
Recognize and differentiate each layer of the function until you arrive at the complete derivative, ensuring each piece is precisely calculated and combined in your final answer.
