Problem 230 - Calculus Volume 1

Question

Find \(\frac{d y}{d x}\) for each function. $$ y=\cos ^{3}(\pi x) $$

Step-by-Step Solution

Answer

The derivative \( \frac{dy}{dx} = -3\pi \cos^2(\pi x) \sin(\pi x) \).

Step 1: Identify the Outer Function

Observe the function given: \( y = \cos^3(\pi x) \). The outer function is \( u^3 \), where \( u = \cos(\pi x) \). This implies we need to find the derivative of \( u^3 \) with respect to \( u \).

Step 2: Differentiate the Outer Function

Using the power rule, the derivative of \( u^3 \) with respect to \( u \) is \( 3u^2 \). Therefore, \( \frac{d}{du} \left( \cos^3(\pi x) \right) = 3\cos^2(\pi x) \).

Step 3: Identify and Differentiate the Inner Function

The inner function is \( \cos(\pi x) \). To differentiate this with respect to \( x \), apply the chain rule:\( \frac{d}{dx} \cos(\pi x) = \frac{d}{d(\pi x)} \cos(\pi x) \times \frac{d}{dx}(\pi x) = -\sin(\pi x) \cdot \pi \).

Step 4: Apply the Chain Rule

By the chain rule, the derivative of \( y = (\cos(\pi x))^3 \) with respect to \( x \) is computed as the derivative of the outer function times the derivative of the inner function. Thus: \( \frac{dy}{dx} = 3\cos^2(\pi x) \cdot (-\sin(\pi x) \times \pi) \).

Step 5: Simplify the Expression

Combine the results from the chain rule. \( \frac{dy}{dx} = -3\pi \cos^2(\pi x) \sin(\pi x) \).

Key Concepts

Chain Rule

The chain rule is a crucial tool in calculus, used when differentiating composite functions. A composite function is essentially a function within another function, like nesting Russian dolls. Say you have functions, like \( f(x) \) and \( g(x) \), combined as \( f(g(x)) \). To differentiate this, we use the chain rule which states that the derivative of \( f(g(x)) \) with respect to \( x \) is the derivative of \( f \) with respect to \( g \) multiplied by the derivative of \( g \) with respect to \( x \).
This rule is especially useful when dealing with nested functions, as it allows you to break down the problem and tackle each layer step by step.

• Identify the outer function and take its derivative.
• Find the inner function and differentiate it with respect to \( x \).
• Multiply these two results together to get the derivative of the entire composite function.

In our example,\( y = \cos^3(\pi x) \), the outer function is \( u^3 \), and the inner function is \( u = \cos(\pi x) \). By applying the chain rule, we solve this differentiation by multiplying the derivatives of these segments.

Power Rule

The power rule is one of the most straightforward differentiation rules. It's applied when differentiating expressions like \( x^n \), where \( n \) is a constant. The formula is simple: bring down the exponent as a coefficient and then reduce the exponent by one, which mathematically is represented as \( \frac{d}{dx}(x^n) = nx^{n-1} \).
In practice, this rule turns differentiation into a quick and easy task, often forming the backbone of more complex operations involving calculus. Its simplicity makes it a favorite in calculus for dealing with polynomial terms.In the exercise provided, we apply the power rule on the outer function \( \cos^3(\pi x) \). We identify the expression in terms of \( u \) as \( u^3 \), bringing down the 3 as a coefficient and reducing the power of \( u \) by one, resulting in \( 3u^2 \). This transformation makes the differentiation process more manageable and paves the way for further applying the chain rule.

Trigonometric Functions

Trigonometric functions are essential in calculus, describing ratios in right triangles and modeling periodic phenomena. They include functions such as sine, cosine, and tangent, each having unique derivatives vital for solving problems in mathematics.

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \(-\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

Understanding these derivatives is key, as they occur frequently in calculus. In our exercise, the function \( \cos(\pi x) \) is differentiated as part of the chain rule application. We note the term's inner derivative, \( \pi x \), and apply the chain rule accordingly. Differentiating \( \cos(\pi x) \) initially gives \(-\sin(\pi x)\), and then it's multiplied by \( \pi \) due to the inner function \( \pi x \).
Mastering these core derivatives and transformations is crucial for tackling a vast array of calculus problems efficiently.