Math Problem Statement
Solution
To find the derivative of the function , we need to use the chain rule. Here’s the step-by-step solution:
Step-by-Step Solution
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Identify the outer and inner functions:
- The outer function is where .
- The inner function is .
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Differentiate the outer function with respect to :
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Differentiate the inner function with respect to using the product rule:
- Here, (using the chain rule again).
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Combine using the chain rule:
- Now we apply the chain rule: .
Thus, the derivative of the function is:
Would you like more details on this solution or have any questions?
Related Questions:
- How do you apply the product rule for derivatives?
- What are the steps to apply the chain rule to composite functions?
- How does the derivative of differ from that of ?
- What is the significance of using the chain rule in differentiation?
- Can you show an example of differentiating a function with a nested exponent?
Tip:
When differentiating exponential functions with a complex exponent, breaking it down using the chain rule makes it much easier to handle.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Exponential Functions
Chain Rule
Product Rule
Formulas
Chain Rule: (f(g(t)))' = f'(g(t)) * g'(t)
Product Rule: (uv)' = u'v + uv'
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Grades 11-12 or early university