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Chapter 8: Problem 15

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

Short Answer

Expert verified
\[-\csc^2 t\]

Step by step solution

01

- Identify the derivative formula for cotangent

Recall the derivative formula for the cotangent function. The derivative of \(\cot t\) is \(-\csc^2 t\).
02

- Apply the derivative formula

Differentiate the given function \(f(t) = \cot t\) using the identified formula. Thus, \frac{d}{dt} [\cot t] = -\csc^2 t\.

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Key Concepts

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

cotangent
Cotangent, represented as \(\text{cot} t\), is a trigonometric function. It is the reciprocal of the tangent function. To understand it, remember that \(\text{cot} t = \frac{1}{\tan t}\). This means that if you know the value of tangent, the value of cotangent will be one divided by tangent. For example, if \[ \tan x = 2, \] then \[ \text{cot} x = \frac{1}{2}. \] This idea makes it easier to comprehend the relationships between different trigonometric functions. Cotangent is periodic, repeating every \pi\ units, just like sine and cosine functions.
trigonometric functions
Trigonometric functions include sine (\text{sin}), cosine (\text{cos}), tangent (\text{tan}), cotangent (\text{cot}), secant (\text{sec}), and cosecant (\text{csc}). These functions are essential in understanding relationships in a right-angled triangle. They also describe periodic phenomena such as sound and light waves. Whether you measure angles in degrees or radians, these functions help in solving various real-life applications. It's key to remember the basic identities such as \[ \text{sin}^2 x + \text{cos}^2 x = 1 \] an invaluable bridge between these functions.
derivative formulas
Derivative formulas are crucial in calculus, helping to find the rate of change of a function. For trigonometric functions, specific derivatives must be memorized. For example, \[ \frac{d}{dt} [\text{sin} t] = \text{cos} t \] and \[ \frac{d}{dt} [\text{cos} t] = -\text{sin} t. \] Importantly, the derivative of cotangent: \[ \frac{d}{dt} [\text{cot} t] = -\text{csc}^2 t. \] When taking derivatives, always ensure you apply the correct formula for the trigonometric function in question to avoid errors.
mathematical notation
Mathematical notation may seem daunting at first glance, but it's simply a way of writing equations and expressions in a concise manner. For instance, differentiation is often symbolized by \[ \frac{d}{dt} \] for derivatives with respect to time, or \[ \frac{d}{dx} \] with respect to a variable \( x \). Parentheses, brackets, and braces are used to clarify the order of operations. Familiarizing yourself with common notations, like \[ f'(t) \] for the derivative of function \( f \), will make understanding advanced concepts much more manageable. Don't hesitate to refer back to basic rules frequently.

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