Math Problem Statement
Differenciate
Solution
The image presents several functions to differentiate. Let's go through each one systematically and find the derivatives.
(a)
The derivative is:
(b)
Using the product rule:
(c)
Using the product rule:
(d)
This is another product of functions, and we can treat as a single term for differentiation: Using the product rule for : Thus:
(e)
This is a quotient, so we use the quotient rule: Simplifying each term:
(f)
This is another quotient, so we apply the quotient rule:
(g)
Using the chain rule:
(h)
Using the chain rule:
(i)
Rewrite as , then use the chain rule:
(j)
Using the chain rule:
(k)
This requires both the chain rule and the power rule:
(l)
Differentiate each term separately:
Would you like any further clarifications on these derivatives?
5 Relative Questions:
- How does the product rule work in differentiating composite functions?
- Can you explain the chain rule used in and ?
- What is the quotient rule and how is it applied to rational functions like in (e)?
- How would you differentiate a function with a trigonometric term raised to a power, as in (k)?
- What is the significance of simplifying the derivative after applying the quotient rule?
Tip:
When differentiating composite functions, always consider the order of operations to apply the chain rule or product rule accurately.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Product Rule
Quotient Rule
Chain Rule
Trigonometric Functions
Formulas
Product Rule: d(uv)/dx = u'v + uv'
Quotient Rule: d(u/v)/dx = (v u' - u v') / v^2
Chain Rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
Theorems
Basic Derivative Rules
Trigonometric Derivatives
Chain Rule
Product Rule
Quotient Rule
Suitable Grade Level
Undergraduate or Advanced High School (Calculus Level)
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