Math Problem Statement
Given f(x)=4/(5-x)^2 find f'(2)
Solution
To find , we first need to differentiate the function .
Step 1: Rewrite the function
We can rewrite as:
Step 2: Differentiate using the chain rule
Now, apply the chain rule to differentiate . The chain rule states that if you have a composite function, the derivative is:
Here, and . First, differentiate the outer function and then multiply by the derivative of the inner function .
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Differentiate :
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Simplify:
Step 3: Substitute
Now, substitute into the derivative:
Final Answer:
Would you like further details or clarification?
Here are five related questions:
- How is the chain rule applied in differentiation?
- What is the power rule for differentiation?
- Can we use an alternative method like implicit differentiation here?
- What is the geometric interpretation of ?
- How do derivatives behave for rational functions?
Tip: Always simplify the function before differentiating to make the process easier!
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Chain Rule
Formulas
Chain Rule: d/dx [g(h(x))] = g'(h(x)) * h'(x)
Power Rule for Differentiation: d/dx [x^n] = n * x^(n-1)
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12