Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(D_{x} y\). $$ y=\cos ^{3} x $$

Step-by-Step Solution

Answer

\(D_{x} y = -3(\cos x)^2 \sin x\).

Step 1: Identify the Chain Rule

To differentiate the expression, we'll need to use the chain rule because the function involves a composition of functions. The chain rule states that if you have a composite function \(y = (g(x))^n\), the derivative is given by \(D_{x}y = n(g(x))^{n-1} \cdot g'(x)\). Here, \(g(x) = \cos x\) and \(n = 3\).

Step 2: Differentiate the Outer Function

First, consider \(y = (\cos x)^3\). According to the chain rule, the derivative of the outside \((g(x))^3\) is \(3(g(x))^2\). So, \(3(\cos x)^2\).

Step 3: Differentiate the Inner Function

Next, differentiate the inner function \(\cos x\) with respect to \(x\). The derivative of \(\cos x\) is \(-\sin x\).

Step 4: Apply the Chain Rule

Combine the results from Steps 2 and 3. According to the chain rule, multiply the derivative of the outer function by the derivative of the inner function: \(D_{x}y = 3(\cos x)^2 \cdot (-\sin x)\).

Step 5: Simplify the Expression

Simplify the expression by multiplying the terms: \(D_{x}y = -3(\cos x)^2 \sin x\).

Key Concepts

Chain Rule

The Chain Rule is a vital tool in calculus, especially when dealing with composite functions. A composite function is essentially a function within a function. For example, consider a function like

• \( y = g(f(x)) \).

To find its derivative, you can't just differentiate one part—you need to account for both layers. The Chain Rule formula is:

• \( \frac{d}{dx}[g(f(x))] = g'(f(x)) \cdot f'(x) \).

In simpler terms, it means to first differentiate the outer function, then multiply by the derivative of the inner function. The Chain Rule helps in simplifying what could otherwise be a complex differentiation problem by breaking it into manageable steps. A common mistake is to apply the rule partially, so always remember: outer times the derivative of the inner!

Derivative

Derivatives are the foundation of calculus, capturing the idea of change. If you have a curve defined as a function, the derivative at any point shows how fast the function value is changing at that point. Formally, the derivative of a function \( f(x) \) is defined as:

• \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \).

The derivative tells you the slope of the tangent line to the function at any point \( x \). This is incredibly helpful in understanding the behavior of functions—whether they are increasing, decreasing, or have reached a maximum or minimum point. Derivatives are used in various applications, from physics to engineering, to help model real-world phenomena.

Trigonometric Functions

Trigonometric functions are those functions that relate to angles and sides of triangles. The primary ones are sine (

• \( \sin \)

) and cosine (

• \( \cos \)

), along with tangent, secant, cosecant, and cotangent. In calculus, these functions are key in modeling periodic phenomena. Differentiating trigonometric functions is standard protocol:

• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \( -\sin x \).

Understanding the derivatives of trigonometric functions allows us to predict how these waves change and behave over time. They are often used in physics and engineering to represent oscillations and waves.

Differentiation

Differentiation is the process of finding a derivative. This operation is about understanding the rate at which a function changes at any point. You apply specific rules, like the power rule, product rule, quotient rule, and chain rule, to differentiate functions. Differentiation allows mathematicians and scientists to

• calculate speed,
• acceleration,
• or any rate of change in a given function.

When you differentiate successfully, you can plot the slope of a function at any time. This is crucial in understanding critical points—where functions reach maximum or minimum values—and in sketching graphs accurately. It simplifies complex mathematical problems by providing precise slopes and rates of change. Differentiation also underpins many other calculus operations, such as integration.