Math Problem Statement
Answer them and make them more simple and get straight to the point while still using limit theorem, lets start again from number 7
Solution
Let's start by solving problem number 7 using the limit definition of derivatives. The function given is:
Step 1: Apply the limit definition of the derivative
The derivative of is:
Step 2: Use product rule for derivatives
We need to use the product rule because is a product of two functions, and . The product rule states:
[ f'(x) = \frac{d}{dx} [\cos(5x) \cdot \tan(5x)] = \cos(5x) \cdot \frac{d}{dx}[\tan(5x)] + \tan(5x) \cdot \frac{d}{dx}[\cos(5x)] ]
Step 3: Derivatives of individual terms
- The derivative of is .
- The derivative of is .
Step 4: Substitute the derivatives back into the product rule
Step 5: Simplify
- , so the first term becomes .
- The second term simplifies to .
Thus, the final expression for the derivative is:
Would you like more details or have any questions?
Here are 5 related questions to consider:
- How do you apply the limit definition of the derivative in simple functions?
- What is the product rule, and how does it simplify differentiating product functions?
- How do we differentiate trigonometric functions like and ?
- Can you simplify trigonometric expressions involving and ?
- How do you handle more complex functions in differentiation?
Tip: Always look for opportunities to apply the product and chain rules when differentiating complex expressions.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Trigonometric Functions
Product Rule
Chain Rule
Formulas
f'(x) = lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
Product Rule: (fg)' = f'g + fg'
Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x)
Theorems
Limit Theorem
Product Rule
Chain Rule
Suitable Grade Level
Grade 11-12 (Advanced Calculus)
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