Question

Finding a Derivative of a Trigonometric Function. In Exercises \(39-54,\) find the derivative of the trigonometric function. $$ y=x+\cot x $$

Short answer

The derivative of \(y = x + \cot x\) is \(\frac{dy}{dx} = 1 - \csc^2x\).

Step-by-step solution

Identify Function Types

In \(y = x + \cot x\), the function has two parts to differentiate: a polynomial part \(x\) and a trigonometric part \(\cot x\). Both of these will be handled individually using the rules of differentiation.

Differentiate the Polynomial Part

The derivative of the polynomial part is given by \(\frac{d}{dx}(x) = 1\). This is because the power rule for derivatives states that the derivative of \(x^n\) is \(nx^{n-1}\), and here \(n = 1\).

Differentiating the Trigonometric Part

The derivative of \(\cot x\) is \(-\csc^2x\). This is a standard rule of differentiation for trigonometric functions. So, \(\frac{d}{dx}(\cot x) = -\csc^2x\).

Combine the Results

The final step is to combine the derivatives of the polynomial and trigonometric parts calculated in steps 2 and 3. This will yield the derivative for the entire function \(y = x + \cot x\). This is done by adding the two results together. So, \(\frac{dy}{dx} = 1 - \csc^2x\).

Key concepts

Understanding Trigonometric Functions

Trigonometric functions are an essential part of calculus, particularly when dealing with angles and periodic phenomena. These functions include the sine, cosine, tangent, cotangent, secant, and cosecant. Each of these functions has its unique characteristics and relationships.

• The cotangent (\(\cot x\)) is the reciprocal of the tangent function, meaning it equals the cosine divided by the sine.
• The cosecant (\(\csc x\)) is the reciprocal of the sine function.

These functions are essential when differentiating equations involving angles and curves. They help us understand how things like waves and rotations work in real-world applications. Exploring these functions in derivatives allows us to see how their rates of change behave.

Applying Differentiation Rules

Differentiation is the process of finding a derivative, which tells us the rate of change of a function. There are several rules that make differentiation systematic and simpler:

• The Power Rule: For any power of \(x\), the derivative is found by multiplying by the exponent and reducing the exponent by one.
• The Sum Rule: This states that the derivative of a sum of functions is simply the sum of their derivatives.

For instance, if we have a function like \(y = x + \cot x\), we differentiate each part. The rules give us a structured way to tackle complex functions by breaking them into manageable parts.

Polynomial Differentiation Explained

Polynomial differentiation revolves around applying the power rule to each term in the polynomial. Consider a function \(f(x) = x^n\), where \(n\) is a constant. The derivative would be \(nx^{n-1}\).

In our specific problem, we have \(x\), which is effectively \(x^1\). Applying the power rule, we differentiate it to \(1\). This direct approach makes polynomials easy to handle, as they break down to simple arithmetic operations. It’s essential to recognize patterns in polynomials to differentiate them quickly.

Derivative of Cotangent

The derivative of the cotangent function requires memorization of standard results or understanding through fundamental trigonometric identities. The cotangent function is defined as \(\cot x = \frac{\cos x}{\sin x}\).

When differentiating \(\cot x\), the result is \(-\csc^2 x\). Here's a hint on why: using the quotient rule to differentiate \(\cot x\), knowing that \(\csc x = \frac{1}{\sin x}\), helps illustrate why the derivative ends up involving the cosecant squared. It's a vital part of differentiating trigonometric functions, contributing to many fields, from engineering to physics.

Exploring the Cosecant Function

The cosecant function \(\csc x\) is one of the lesser-known trigonometric functions, but it's quite important. As the reciprocal of the sine function, it’s defined as \(\csc x = \frac{1}{\sin x}\).

• Its graph shows vertical asymptotes where the sine is zero, reflecting its undefined nature at those points.
• The derivative, \(-\csc x \cot x\), emerges when exploring how trigonometric functions' rates of change need careful handling.

Understanding \(\csc x\) helps demystify its behavior, especially when interlinked with other trigonometric functions in differentiation, emphasizing the intricate dance of rates within calculus.