Math Problem Statement
Solution
The image contains a set of algebraic functions for which you are asked to find the first derivative. Let's solve each problem step by step:
1.
To find the first derivative, use the power rule .
2.
Here, we need to use the chain rule. Let's break it down step by step.
Let , so .
First, differentiate with respect to :
Next, differentiate with respect to :
Now, apply the chain rule:
Thus, the derivative is:
3.
This simplifies to since the numerator and denominator are the same, except for the case when , where it is undefined.
For , the derivative is:
4.
Use the product rule , where and .
First, find and :
Now apply the product rule:
Do you have any questions or would you like further clarification on any of these solutions?
5 Related Questions:
- What is the chain rule, and how is it applied to composite functions?
- How does the power rule for derivatives apply to terms like ?
- Can you explain the product rule and when it's necessary?
- How do you find the derivative of a square root function?
- Why does the derivative of a constant function equal zero?
Tip:
When dealing with products and compositions of functions, always consider whether you need to use the product rule, chain rule, or both together!
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Math Problem Analysis
Mathematical Concepts
Derivatives
Power Rule
Chain Rule
Product Rule
Formulas
Power Rule: d/dx(x^n) = n*x^(n-1)
Chain Rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)
Product Rule: d/dx[uv] = u'v + uv'
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Undergraduate
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