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In Exercises $1-18,$ find $d y / d x.$ $$y=\frac{\cos x}{x}+\frac{x}{\cos x}$$

   In Exercises $1-18,$ find $d y / d x.$
$$y=\frac{\cos x}{x}+\frac{x}{\cos x}$$

Calculus

George B. Thomas,… 12th Edition

Chapter 3, Problem 14

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Solved by Expert Patrick Vaughn

07/31/2020

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These are $\frac{\cos x}{x}$ and $\frac{x}{\cos x}$. Show more…

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In Exercises $1-18,$ find $d y / d x.$ $$y=\frac{\cos x}{x}+\frac{x}{\cos x}$$

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Key Concepts

Sum Rule

The sum rule is a fundamental principle in differentiation which states that the derivative of a sum of functions is equal to the sum of their individual derivatives. This rule allows us to differentiate expressions term-by-term, simplifying the process when dealing with multiple functions added together.

Quotient Rule

The quotient rule is used when differentiating a function that is written as the ratio of two functions. It provides a formula to calculate the derivative of a quotient, specifically stating that the derivative is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Chain Rule

The chain rule is a key technique in differentiation that deals with composite functions. It is used to differentiate a function whose variable is itself another function. The chain rule states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

Derivatives of Trigonometric Functions

Understanding the derivatives of trigonometric functions is essential for differentiating expressions involving trigonometric terms. For example, knowing that the derivative of cosine is negative sine allows you to differentiate functions involving cosine and other trigonometric components correctly.

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