Problem 18 - Calculus Early Transcendentals

Question

Find \(f^{\prime}(x)\) $$ f(x)=4 x+5 \sin ^{4} x $$

Step-by-Step Solution

Answer

The derivative is \(f'(x) = 4 + 20 \sin^3(x) \cos(x)\).

Step 1: Identify the Components of the Function

The function given is \( f(x) = 4x + 5 \sin^4(x) \). It consists of two parts: a linear term \(4x\) and a trigonometric term \(5 \sin^4(x)\). To find the derivative, we will differentiate each part individually.

Step 2: Differentiate the Linear Term

The first term \(4x\) is a linear function. The derivative of \(ax\) with respect to \(x\) is simply \(a\). Therefore, the derivative of \(4x\) is \(4\).

Step 3: Differentiate the Trigonometric Term

The term \(5 \sin^4(x)\) is a composition of functions, specifically a power function and a trigonometric function. We apply the chain rule here. Differentiate the outer function \(y^4\) where \(y = \sin(x)\), which yields \(4y^3\), then multiply by the derivative of the inner function, \(\sin(x)\), which is \(\cos(x)\). Thus, the derivative of \(\sin^4(x)\) is \(4\sin^3(x) \cos(x)\). Multiply this by \(5\) to get \(20\sin^3(x) \cos(x)\).

Step 4: Combine the Derivatives

Combine the derivatives of both terms obtained from Step 2 and Step 3. The derivative \(f'(x)\) is the sum of these derivatives: \(f'(x) = 4 + 20 \sin^3(x) \cos(x)\).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions and is expressed in the form \(ax + b\), where \(a\) and \(b\) are constants. It has a constant rate of change, making it a straight line when graphed. The derivative of a linear function is straightforward. Suppose you have a function \(f(x) = 4x\). The derivative, which represents the slope of the function, will constantly be \(a\).

• Thus, for \(f(x) = 4x\), the derivative is \(f'(x) = 4\).
• This means the slope or the rate of change of the function \(4x\) is 4.

The critical aspect to remember is that derivatives of linear functions yield the coefficient of \(x\). This makes calculating the derivatives of linear terms highly straightforward.

Trigonometric Functions

Trigonometric functions like sine and cosine are periodic functions, which are crucial in calculus, especially when dealing with oscillations and waves. In our exercise, we encountered the function \(5 \sin^4(x)\). This is a composition involving a trigonometric function with a power.

To differentiate \(\sin^4(x)\), recognize it as \((\sin(x))^4\). This interpretation allows us to use rules associated with powers and trigonometric functions.

• The derivative of \(\sin(x)\) is \(\cos(x)\).
• When you have a sine function raised to a power, you first apply the power rule: bring down the power and multiply it by the sine function raised to one less than the power.

Therefore, the derivative of \(\sin^4(x)\) becomes \(4 \sin^3(x) \cdot \cos(x)\). It's crucial to remember the periodic nature and basic derivative formulas of trigonometric functions when working through calculus problems.

Chain Rule

The chain rule is an essential differentiation rule in calculus, used when differentiating compositions of functions. This becomes especially relevant when functions are nested within each other, like in our trigonometric example \(5 \sin^4(x)\).

The chain rule states that to differentiate a composite function \(f(g(x))\), multiply the derivative of the outer function by the derivative of the inner function:

• If \(y = g(x)\), first find \(f'(y)\).
• Then multiply by \(g'(x)\).

In our exercise, ending with \(\sin^4(x)\):

• Identify the inner function \(\sin(x)\).
• Find its derivative, \(\cos(x)\).
• Then, for the outer function \((y^4)\), the derivative is \(4y^3\).
• Combine them: \(4 \sin^3(x) \cdot \cos(x)\).

By applying the chain rule, you correctly account for the rate of change of the nested functions, leading to accurate derivative results. Mastery of the chain rule is pivotal for success in calculus as it frequently appears in complex function differentiation.