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Chapter 8: Problem 16

Differentiate (with respect to \(t\) or \(x\) ): $$f(t)=\cot 3 t$$

Short Answer

Expert verified
The derivative of \( f(t) = \cot(3t) \) is \( f'(t) = -3 \csc^2(3t) \).

Step by step solution

01

Identify the Function to Differentiate

Given function: \[ f(t) = \cot 3t \] The task is to find the derivative of this function with respect to \(t\).
02

Recall the Derivative of the Cotangent Function

Recall that the derivative of \(\cot u\) with respect to \(u\) is \[ \frac{d}{du} (\cot u) = -\csc^2 u \] where \(\csc u\) is the cosecant function.
03

Apply the Chain Rule

The function \( f(t) \) can be written as \[ f(t) = \cot(3t) \] Using the chain rule for differentiation, we have: \[ \frac{d}{dt} [\cot(3t)] = \frac{d}{d(3t)} [\cot(3t)] \cdot \frac{d(3t)}{dt} \]
04

Differentiate the Cotangent Function

First, differentiate \( \cot(3t)\) with respect to \(3t\). Using the rule from Step 2, we get: \[ \frac{d}{d(3t)} [\cot(3t)] = -\csc^2(3t) \]
05

Differentiate the Inner Function

Now differentiate the inner function, \(3t\), with respect to \(t\): \[ \frac{d(3t)}{dt} = 3 \]
06

Combine the Results

Combining the results from Steps 4 and 5, we get: \[ \frac{d}{dt} [\cot(3t)] = -\csc^2(3t) \cdot 3 \] Therefore, \[ f'(t) = -3 \csc^2(3t) \]

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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 tool in calculus for finding the derivative of composite functions. When you have a function within another function, like in our problem with \(f(t)\) being \(\cot(3t)\), you'll use the chain rule to differentiate it. The rule states that if \(y = g(h(x))\), then the derivative of \(y\) with respect to \(x\) is the product of the derivative of \(g\) with respect to \(h\) and the derivative of \(h\) with respect to \(x\).
Derivative of Cotangent
The derivative of the cotangent function \(\cot(x)\) is essential for solving our problem. Recall from standard calculus that the derivative of \(\cot(u)\) with respect to \(u\) is \(-\csc^2(u)\). The cosecant function \(\csc(u)\) itself is the reciprocal of the sine function, i.e., \(\csc(u) = 1/\sin(u)\). Knowing this helps us understand why the derivative is \(-\csc^2(u)\). It shows up frequently in trigonometric differentiation tasks.
Cosecant Function
The cosecant function, abbreviated as \(\csc(x)\), is the reciprocal of the sine function, meaning \(\csc(x) = 1/\sin(x)\). It is important to be comfortable with this function when dealing with derivatives involving the cotangent. This function often appears in the derivatives of trigonometric functions, especially in the context of higher order derivatives and composite functions. It makes the derived expressions more complex but also very insightful in understanding periodic properties.

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Most popular questions from this chapter

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