Problem 20 - Calculus

Question

Find the derivatives of the functions in \(1-26\). $$ \frac{\sin x-\cos x}{\sin x+\cos x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{2\sin(2x)}{(\sin x + \cos x)^2} \).

Step 1: Identify the Function

The function given is \( f(x) = \frac{\sin x - \cos x}{\sin x + \cos x} \). Notice that this function is a quotient of two trigonometric expressions.

Step 2: Recall Derivative of a Quotient Rule

To find the derivative of a quotient \( \frac{u}{v} \), use the formula: \[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]. Here, \( u(x) = \sin x - \cos x \) and \( v(x) = \sin x + \cos x \).

Step 3: Differentiate the Numerator

Differentiate \( u(x) = \sin x - \cos x \). The derivative is \( u'(x) = \cos x + \sin x \), because the derivative of \( \sin x \) is \( \cos x \) and the derivative of \( -\cos x \) is \( \sin x \).

Step 4: Differentiate the Denominator

Differentiate \( v(x) = \sin x + \cos x \). The derivative is \( v'(x) = \cos x - \sin x \), because the derivative of \( \sin x \) is \( \cos x \) and the derivative of \( \cos x \) is \( -\sin x \).

Step 5: Apply the Quotient Rule

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

Step 6: Simplify the Expression

Compute the expression: First, expand each term in the numerator: \((\cos x + \sin x)(\sin x + \cos x) = \cos^2 x + \cos x \sin x + \sin x \cos x + \sin^2 x = 1 + 2\sin x \cos x = 1 + \sin(2x)\). Similarly, \((\sin x - \cos x)(\cos x - \sin x) = \sin^2 x - \sin x \cos x - \cos x \sin x + \cos^2 x = 1 - \sin(2x)\). Thus, the numerator becomes \(1 + \sin(2x) - (1 - \sin(2x)) = 2\sin(2x)\).

Step 7: Final Expression

The derivative simplifies to \( f'(x) = \frac{2\sin(2x)}{(\sin x + \cos x)^2} \). This is the simplified expression for the derivative of the function.

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus used to find the derivative of a division of two functions with respect to a variable. When you have a function that can be expressed as a quotient, as seen in the form \( f(x) = \frac{u(x)}{v(x)} \), where both \( u(x) \) and \( v(x) \) are differentiable, the derivative \( f'(x) \) can be computed using the quotient rule:

• Identify the numerator \( u(x) \) and the denominator \( v(x) \).
• Calculate the derivative of the numerator: \( u'(x) \).
• Calculate the derivative of the denominator: \( v'(x) \).
• Apply the quotient rule formula: \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \].

After finding these derivatives, substitute them back into the quotient rule formula. This approach is instrumental in understanding how each term of the quotient influences the overall shape and slope of the curve defined by the quotient function.

Trigonometric Functions

Trigonometric functions like \( \sin x \) and \( \cos x \) are periodic functions fundamental in mathematics, especially in calculus. They model cycles, waves, and oscillations, which are prevalent in both pure and applied sciences.

• The basic trigonometric functions are \( \sin x \) and \( \cos x \).
• The derivatives of these functions are critical to remember:
• The derivative of \( \sin x \) is \( \cos x \).
• The derivative of \( \cos x \) is \(-\sin x \).
• These derivatives illustrate how rates of change for these functions relate directly to the other, showing their intrinsic link.

Understanding these derivatives will help when performing tasks like using the quotient rule, as in this exercise, or analyzing motion in physics, among other applications. The interrelationship between \( \sin \) and \( \cos \), including their derivatives, forms the basis of many calculus operations and transformations.

Simplification of Expressions

Simplifying expressions is a crucial step in finding derivatives and understanding calculus comprehensively. This process involves combining like terms, factoring, expanding, and reducing expressions to their simplest form to make them easier to interpret and solve.

• The aim is to identify terms that can be simplified or reduced.
• Look for opportunities to factor or combine terms, such as using angle sum identities in trigonometry.
• Always check for identity relationships or algebraic tricks like \( \sin(2x) = 2\sin x\cos x \).

For instance, in our exercise concerning the derivative of the function \( \frac{\sin x - \cos x}{\sin x + \cos x} \), simplification helps reduce the derivative to the expression \( \frac{2\sin(2x)}{(\sin x + \cos x)^2} \). This not only makes the representation cleaner but also provides deeper insight into the behavior and characteristics of the function.