Problem 5
Question
Find \(f^{\prime}(x)\). $$f(x)=\frac{5-\cos x}{5+\sin x}$$
Step-by-Step Solution
Verified Answer
\( f'(x) = \frac{1 + 5(\sin x - \cos x)}{(5 + \sin x)^2} \)
1Step 1: Identify the Function Type
The given function \( f(x) = \frac{5 - \cos x}{5 + \sin x} \) is a quotient of two functions. This suggests we'll use the Quotient Rule to find its derivative.
2Step 2: Recall the Quotient Rule Formula
The Quotient Rule states that for two functions \( u(x) \) and \( v(x) \), the derivative of their quotient \( \frac{u(x)}{v(x)} \) is given by:\[ \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \]
3Step 3: Identify \( u(x) \) and \( v(x) \)
From \( f(x) = \frac{5 - \cos x}{5 + \sin x} \), assign\[ u(x) = 5 - \cos x \] \[ v(x) = 5 + \sin x \]
4Step 4: Differentiate \( u(x) \) and \( v(x) \)
Find the derivatives of \( u(x) \) and \( v(x) \):- Derivative of \( u(x) = 5 - \cos x \) is \( u'(x) = \sin x \).- Derivative of \( v(x) = 5 + \sin x \) is \( v'(x) = \cos x \).
5Step 5: Apply the Quotient Rule Formula
Substitute \( u(x) \), \( v(x) \), \( u'(x) \), and \( v'(x) \) into the Quotient Rule formula:\[ f'(x) = \frac{(5 + \sin x)(\sin x) - (5 - \cos x)(\cos x)}{(5 + \sin x)^2} \]
6Step 6: Simplify the Expression
Expand and simplify the expression:- The numerator becomes: \[ \sin x(5 + \sin x) - \cos x(5 - \cos x) = 5 \sin x + \sin^2 x - 5 \cos x + \cos^2 x \]- Use the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to simplify: \[ = 5 \sin x + 1 - 5 \cos x \]- The numerator is simplified to: \[ = 1 + 5(\sin x - \cos x) \]- Therefore, \( f'(x) = \frac{1 + 5(\sin x - \cos x)}{(5 + \sin x)^2} \).
Key Concepts
DerivativesTrigonometric FunctionsSimplification
Derivatives
Understanding derivatives is key to solving calculus problems, especially when dealing with complex functions. A derivative measures how a function changes as its input varies. It is often described as the function's "instantaneous rate of change" and can be visualized as the slope of the tangent to the curve at any given point.
To find the derivative of more intricate functions, like a quotient of two functions, specific rules are applied, such as the Quotient Rule.
Learning these rules can simplify finding the derivative, showing patterns and connections that are not immediately apparent. Working through derivatives systematically by identifying function types, like quotients, and applying appropriate rules, helps manage the complexity of calculus.
To find the derivative of more intricate functions, like a quotient of two functions, specific rules are applied, such as the Quotient Rule.
Learning these rules can simplify finding the derivative, showing patterns and connections that are not immediately apparent. Working through derivatives systematically by identifying function types, like quotients, and applying appropriate rules, helps manage the complexity of calculus.
Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental in understanding waves and oscillations. These functions are periodic and provide a powerful means to describe cyclical phenomena in calculus.
In our exercise, the trigonometric functions \(\sin x\) and \(\cos x\) appear in both the numerator and the denominator. These functions have specific derivatives. For instance, the derivative of \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\).
Handling trigonometric derivatives requires familiarity with these derivations and knowledge of identities, such as the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), which simplifies expressions in calculus problems.
In our exercise, the trigonometric functions \(\sin x\) and \(\cos x\) appear in both the numerator and the denominator. These functions have specific derivatives. For instance, the derivative of \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\).
Handling trigonometric derivatives requires familiarity with these derivations and knowledge of identities, such as the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), which simplifies expressions in calculus problems.
Simplification
Simplification in calculus helps make complex derivative expressions more manageable and interpretable. After applying a rule like the Quotient Rule, the initial result can often appear messy, demanding further simplification.
This process involves expanding and combining like terms, as well as applying mathematical identities. For example, the Pythagorean identity helps transition \(\sin^2 x + \cos^2 x\) to just \(1\), dramatically simplifying expressions.
Effective simplification boils down to seeing which terms can be combined or cancelled, often revealing a clearer mathematical expression, making it easier to understand, interpret, and work with. This enhances both computational efficiency and comprehension.
This process involves expanding and combining like terms, as well as applying mathematical identities. For example, the Pythagorean identity helps transition \(\sin^2 x + \cos^2 x\) to just \(1\), dramatically simplifying expressions.
Effective simplification boils down to seeing which terms can be combined or cancelled, often revealing a clearer mathematical expression, making it easier to understand, interpret, and work with. This enhances both computational efficiency and comprehension.
Other exercises in this chapter
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