Problem 71 - Calculus for Biology and Medicine

Question

Find the derivatives of the following functions: $$ f(x)=\sin 2 x+\sin ^{2} x $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 2\cos(2x) + 2\sin(x)\cos(x) \).

Step 1: Identify the derivative rules needed

To find the derivative of the function given, we need to use the standard derivative formulas for sine functions and power functions. Additionally, the chain rule is necessary for composite functions like \( \sin(2x) \) and \( \sin^2(x) \).

Step 2: Derivative of \( \sin(2x) \)

The derivative of \( \sin(2x) \) requires the chain rule. First, the derivative of \( \sin(u) \) is \( \cos(u) \); thus, for \( \sin(2x) \) we replace \( u \) with \( 2x \). The chain rule tells us to multiply by the derivative of \( 2x \), which is 2. Hence, the derivative of \( \sin(2x) \) is \( 2 \cos(2x) \).

Step 3: Derivative of \( \sin^2(x) \)

For \( \sin^2(x) \), we also use the chain rule. The outer function is the squaring function, with the derivative being \( 2 \cdot \sin(x) \) multiplied by the derivative of the inner \( \sin(x) \), which is \( \cos(x) \). Therefore, the derivative of \( \sin^2(x) \) is \( 2 \sin(x) \cos(x) \).

Step 4: Combine derivatives for \( f(x) = \sin(2x) + \sin^2(x) \)

Add the derivatives of each term. From Step 2, we have \( 2\cos(2x) \) and from Step 3, we have \( 2\sin(x)\cos(x) \). Therefore, the derivative \( f'(x) \) is given by the sum: \( f'(x) = 2\cos(2x) + 2\sin(x)\cos(x) \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used to find the derivative of composite functions. A composite function is essentially a function within another function, like \( \sin(2x) \). The rule can be remembered as a two-step process: first, differentiate the outer function, then multiply by the derivative of the inner function.
For example:

• Consider \( \sin(2x) \). Here, \( \sin(u) \) is the outer function, and \( u = 2x \) is the inner function.
• Differentiate \( \sin(u) \) to get \( \cos(u) \).
• Then, multiply by the derivative of \( 2x \), which is 2.
• This results in the derivative \( 2\cos(2x) \).

Using the Chain Rule helps us tackle complicated derivatives, breaking them into manageable parts. After a bit of practice, applying it becomes intuitive!

Trigonometric Functions

Trigonometric functions, like sine and cosine, are essential in calculus due to their periodic nature and wide application in various fields. These involve angles and the ratios formed by the sides of a right triangle.
Let's focus on the sine function:

• Its derivative is \( \cos(x) \).
• This relationship helps us understand how the slope (or rate of change) of the sine curve behaves.

When dealing with derivatives like \( \sin^2(x) \), recognizing these basic derivatives is crucial.
Additionally, knowing the derivatives of these functions aids in solving physics and engineering problems where wave-like motions describe systems or signals.

Power Rule

The Power Rule is one of the simplest and most frequently used derivative rules in calculus. It simplifies the process of differentiating functions in the form of \( x^n \).
Here's how it works:

• If \( f(x) = x^n \), then the derivative, \( f'(x) \), is \( nx^{n-1} \).

Applying this rule to \( \sin^2(x) \):

• Regard \( \sin(x) \) as \( u \) where \( u = \sin(x) \).
• The function rearranges into \( u^2 \).
• Using the Power Rule, differentiate to obtain \( 2u \cdot u' \).
• Substitute back \( \sin(x) \) for \( u \) and multiply by \( \cos(x) \) to account for \( u' \). This gives \( 2\sin(x)\cos(x) \).

The Power Rule makes differentiating polynomial-like expressions straightforward, significantly simplifying the chain-based derivatives of functions.