Question

$47-50$ Find $y^{\prime}$ and $y "$ $$y=\cos ^{2} x$$

   $47-50$ Find $y^{\prime}$ and $y "$
$$y=\cos ^{2} x$$

Calculus Early Transcendentals

Calculus Early Transcendentals

James Stewart 7th Edition

Chapter 3, Problem 48

Instant Answer

Solved by Expert Darshan Maheshwari

10/15/2020

Step 1

We use the chain rule for differentiation, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. Here, the outer function is $u^{2}$ and the inner function is $\cos x$. So, we Show more…

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$47-50$ Find $y^{\prime}$ and $y "$ $$y=\cos ^{2} x$$

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Key Concepts

Second Derivative

The second derivative provides information about the concavity and the acceleration of changes in the function. It is obtained by differentiating the first derivative, using the same differentiation rules, and is significant in understanding the function's curvature and behavior.

Trigonometric Identities: Double Angle Formulas

Double angle formulas, such as expressing products of sine and cosine in terms of sine or cosine of double angles, are used to simplify the derivative expressions. They provide a way to rewrite derivatives in a more streamlined or compact form.

Chain Rule

The chain rule is used to differentiate composite functions. In differentiation, when a function is nested within another function, the derivative is found by taking the derivative of the outer function evaluated at the inner function and multiplying it by the derivative of the inner function.

Power Rule

The power rule states that the derivative of a function raised to a power 'n' is n times the function raised to the power n-1, multiplied by the derivative of the base function. This concept is essential when differentiating expressions like the square of a trigonometric function.

Differentiation of Trigonometric Functions

This concept involves knowing the derivatives of basic trigonometric functions, such as the derivative of cosine being negative sine. These fundamental derivatives are applied, often alongside the chain and power rules, when differentiating functions involving trigonometric expressions.

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