Problem 176 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for the given functions. $$y=3 \csc x+\frac{5}{x}$$

Step-by-Step Solution

Answer

\(\frac{d y}{d x} = -3 \csc x \cot x - \frac{5}{x^2}\).

Step 1: Identify the function and its terms

The function is given as \(y = 3 \csc x + \frac{5}{x}\). We have two separate terms: \(3 \csc x\) and \(\frac{5}{x}\). We'll find the derivative of each term separately.

Step 2: Differentiate the first term

Differentiate \(3 \csc x\) with respect to \(x\). We know the derivative of \(\csc x\) is \(-\csc x \cot x\). Therefore, the derivative of \(3 \csc x\) is \(3(-\csc x \cot x) = -3 \csc x \cot x\).

Step 3: Differentiate the second term

Differentiate \(\frac{5}{x}\) with respect to \(x\). Rewriting \(\frac{5}{x}\) as \(5x^{-1}\), the derivative is found using the power rule: multiply by the power \(-1\) and then decrease the power by 1. So, it becomes \(-5x^{-2}\), which can also be written as \(-\frac{5}{x^2}\).

Step 4: Combine the derivatives

The derivative of the entire function \(y\) is the sum of the derivatives of its parts. Therefore, \(\frac{d y}{d x} = -3 \csc x \cot x - \frac{5}{x^2}\).

Key Concepts

Derivative Calculation

Calculating derivatives is a fundamental aspect of calculus. It involves finding the slope of a function at any given point. In simpler terms, it tells us how the function is changing at a specific point. For the function given in our exercise, the function is split into two terms: \(3 \csc x\) and \(\frac{5}{x}\).
Each term is handled individually to find its derivative, which are then combined to get the overall derivative of the function.
Understanding each step of the derivative calculation is essential for mastering calculus.

Trigonometric Functions

Trigonometric functions, such as \(\csc x\), play a critical role in calculus as they often appear in various types of functions.
The derivative of \(\csc x\) is \(-\csc x \cot x\) and is crucial when you encounter derivatives involving cosecant.
It's important to memorize these derivatives and understand their derivations, as they frequently come up in differentiation tasks.
In our exercise, knowing the derivative of \(\csc x\) allowed us to find the derivative of \(3 \csc x\) quickly by simply multiplying the derivative of \(\csc x\) by 3.

Power Rule

The power rule is one of the simplest and most commonly used rules for differentiation in calculus.
For the power rule, if you have a term such as \(x^n\), its derivative will be \(nx^{n-1}\).
In the exercise, we use this rule to find the derivative of \(\frac{5}{x}\) by rewriting it as \(5x^{-1}\).
Applying the power rule, you multiply by the power (\(-1\)) and then decrease the power by one, which gives us \(-5x^{-2}\) or \(-\frac{5}{x^2}\).
Mastering the power rule allows for quick and efficient differentiation of many algebraic expressions.