Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=1-\cos ^{2} x $$

Step-by-Step Solution

Answer

The derivative is \( \sin(2x) \).

Step 1: Use a Trigonometric Identity

Recognize that the expression \(1 - \cos^2 x\) is a known trigonometric identity. Using the Pythagorean identity, this expression simplifies to \(\sin^2 x\). Therefore, we have \(y = \sin^2 x\).

Step 2: Differentiate Using the Chain Rule

The function \(y = \sin^2 x\) can be rewritten as \((\sin x)^2\). To differentiate a squared function of another function, apply the chain rule. The derivative of \((u)^2\) with respect to \(x\) is \(2u\frac{du}{dx}\). Here, let \(u = \sin x\), so \(y = u^2\).

Step 3: Calculate the Derivative

Using \(u = \sin x\), the derivative \(\frac{du}{dx}\) is \(\cos x\). Apply the chain rule from Step 2: \[D_x(u^2) = 2u\frac{du}{dx} = 2(\sin x)(\cos x)\].

Step 4: Simplify the Expression

The derivative \(D_x y = 2(\sin x)(\cos x)\) can be further simplified using the double angle identity: \(2 \sin x \cos x = \sin(2x)\). Hence, the expression simplifies to \(D_x y = \sin(2x)\).

Key Concepts

chain rule

The chain rule is a fundamental technique in calculus for differentiating composite functions. When a function is inside another function, the chaining of derivatives allows us to differentiate the outer function concerning the inner function. To see why the chain rule is essential, imagine we are differentiating a function like \(( ext{some function of } x)^2\). We need to break it down into manageable bits. Let's say we have \(y = ( ext{inner function})^2\). The derivative would involve two parts:

• The derivative of the outer part, in this case, \(2( ext{inner function})\).
• The derivative of the inner part concerning \(x\). If the inner part is \(u\), then \(\frac{du}{dx}\).

Putting it all together gives us the formula \(D_x(u^2) = 2u\frac{du}{dx}\). This is how the chain rule helps by combining both the derivative of the inner and the outer parts. Once you master applying the chain rule, tackling composite functions becomes a systematic process. Remember to look for one function inside another and apply the chain rule accordingly.

trigonometric identities

Trigonometric identities are equations that involve trigonometric functions and are true for every value of the variables involved. These identities are handy in calculus, especially for simplifying complex trigonometric expressions before differentiating or integrating.In the given problem, we use a trigonometric identity to simplify the expression \(1 - \cos^2x\) to \(\sin^2x\). This is based on the Pythagorean identity:

• \(\sin^2x + \cos^2x = 1\)

Rearranging it gives us \(\sin^2x = 1 - \cos^2x\). Recognizing this simplifies the differentiation problem significantly, reducing what could be a long process into a simple application of the chain rule. Trigonometric identities are invaluable for converting expressions into more workable forms. When you identify such identities, you can handle complex problems more efficiently by breaking them into simpler pieces.

derivative simplification

Derivative simplification involves making a derivative more manageable or converting it into a form that is easier to work with or interpret. After finding the derivative, simplification can reveal alternative forms that offer insight or ease of use—in this case, employing the double angle identity.For the derivative \(2(\sin x)(\cos x)\) we use the double angle formula:

• \(\sin(2x) = 2\sin(x)\cos(x)\)

Using this identity, the derivative \(2(\sin x)(\cos x)\) simplifies to \(\sin(2x)\). Simplifying derivatives isn’t merely for neatness. It often exposes underlying patterns or symmetries that might not be apparent in the expanded form.Understanding how to employ identities to simplify derivatives is a critical skill. It makes further calculus operations easier and often yields more meaningful results. Always check if an identity can lead to a cleaner, more insightful result.