Problem 234 - Calculus Volume 1

Question

For the following exercises, find \(\frac{d y}{d x}\) for each function. $$y=x^{2} \cos ^{4} x$$

Step-by-Step Solution

Answer

\( \frac{d y}{d x} = 2x \cos^4 x - 4x^2 \cos^3 x \sin x \)

Step 1: Identify the Function Type

The function given is a product of two smaller functions: \( u(x) = x^2 \) and \( v(x) = \cos^4 x \). Recognize that this requires the use of the product rule in differentiation.

Step 2: Apply the Product Rule Formula

The product rule states that if \( y = u(x) v(x) \), then the derivative \( \frac{d y}{d x} = u'(x) v(x) + u(x) v'(x) \). We will apply this to our function.

Step 3: Differentiate \( u(x) = x^2 \)

The derivative of \( u(x) = x^2 \) with respect to \( x \) is \( u'(x) = 2x \).

Step 4: Differentiate \( v(x) = \cos^4 x \) Using the Chain Rule

Express \( v(x) = \cos^4 x \) as \( (\cos x)^4 \). Using the chain rule, the derivative is \( v'(x) = 4\cos^3 x (-\sin x) \) or \( v'(x) = -4\cos^3 x \sin x \).

Step 5: Substitute into the Product Rule Formula

Substitute \( u'(x) = 2x \), \( v(x) = \cos^4 x \), \( u(x) = x^2 \), and \( v'(x) = -4\cos^3 x \sin x \) into the product rule formula: \[ \frac{d y}{d x} = (2x)(\cos^4 x) + (x^2)(-4\cos^3 x \sin x) \].

Step 6: Simplify the Expression

Combine and simplify the terms: \[ \frac{d y}{d x} = 2x \cos^4 x - 4x^2 \cos^3 x \sin x \]. This is the expression for \( \frac{d y}{d x} \).

Key Concepts

Differentiation

Differentiation is a key concept in calculus that helps us find how a function's output changes as its input changes. It's essentially about finding the "rate of change" or the "slope" of a function at any given point.

• To differentiate a function, we apply various rules, depending on the form of the function.
• The most basic rule is the power rule, used for differentiating functions of the form \(x^n\), where the derivative is \(nx^{n-1}\).

For example, if we have \(u(x) = x^2\), using the power rule, we find that \(u'(x) = 2x\). This informs us how \(x^2\) changes as \(x\) changes. In our exercise, the function to differentiate involves both the product and power rules.
The derivative helps provide us with valuable insights into the function's behavior, such as identifying points where it is increasing or decreasing.

Chain Rule

The chain rule is a fundamental tool in differentiation when dealing with composite functions. A composite function is essentially a function within another function, like \((\cos x)^4\) found in the exercise.

• The chain rule allows us to differentiate such functions efficiently by breaking them down into their constituent parts.
• The chain rule is mathematically expressed as \(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\).

In the example function \(v(x) = \cos^4 x\), we can rewrite it as \((\cos x)^4\). Applying the chain rule, first differentiate the outer function, treating \(\cos x\) as a single variable to get \(4 \cos^3 x\).
Next, differentiate the inner function \(\cos x\), which gives \(-\sin x\). Multiply these derivatives together: \(v'(x) = 4 \cos^3 x (-\sin x)\) or \(-4\cos^3 x \sin x\). This yields the rate of change for the composite function.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in contexts involving periodic phenomena. The basic trigonometric functions are \(\sin x\), \(\cos x\), and \(\tan x\), each with its unique derivative.

• When differentiating trigonometric functions, it's important to memorize their derivatives: \(\frac{d}{dx}(\sin x) = \cos x\) and \(\frac{d}{dx}(\cos x) = -\sin x\).
• These derivatives help us understand how trigonometric functions behave as their inputs change.

In our exercise, we dealt with \(\cos x\), whose derivative, as part of the chain rule, was used to find \((\cos x)^4\)'s derivative. Recognizing the behavior of \(\cos x\) being stretched (or compressed) multiple times in \((\cos x)^4\) highlights the interplay of components in composite trigonometric functions.
Mastering trigonometric derivatives is essential for handling functions involving angles and cycles, which appear frequently in both theoretical and applied mathematics.