Math Problem Statement
Solution
The given function is:
To find the derivative , we can apply the quotient rule for derivatives, which states:
Here, and .
Step 1: Compute and
Step 2: Apply the Quotient Rule
Substituting into the quotient rule formula:
Step 3: Simplify the Numerator
Expanding both terms in the numerator:
Since , this simplifies to:
Step 4: Combine the Two Terms in the Numerator
[ \text{Numerator} = [(a^2 + b^2) \sin x \cos x - ab] - [(a^2 + b^2) \sin x \cos x + ab] ]
Step 5: Final Derivative
Now, we can write the derivative as:
This is the required derivative of the given function.
Let me know if you would like more details on any of the steps! Here are some related questions:
- How do you apply the quotient rule to a general function?
- What is the physical interpretation of the derivative for trigonometric functions?
- How do we differentiate products of trigonometric functions?
- Can we use the chain rule to derive the result in another way?
- What happens if or becomes zero in this context?
Tip: The quotient rule can simplify complex rational functions but always requires careful application of derivatives for both the numerator and the denominator.
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Math Problem Analysis
Mathematical Concepts
Calculus
Trigonometric Functions
Differentiation
Formulas
Quotient Rule: (d/dx) [u(x)/v(x)] = [v(x)u'(x) - u(x)v'(x)] / [v(x)]^2
Derivative of sin(x): cos(x)
Derivative of cos(x): -sin(x)
Theorems
Quotient Rule
Trigonometric Derivatives
Suitable Grade Level
Grade 12 or College Calculus
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