Math Problem Statement
{x^5}{(x^2+4)^2} find this Different fractions
Solution
To find the derivative of the expression:
we'll use the quotient rule along with the product rule and chain rule as needed.
The quotient rule for differentiation states that:
Here:
- and
Step 1: Differentiate
Step 2: Differentiate using the chain rule
Let , so .
Now, differentiate :
Now apply the chain rule to differentiate :
Step 3: Apply the quotient rule
Now, substitute into the quotient rule formula:
Step 4: Simplify the expression
Simplify the numerator:
We can factor out from both terms in the numerator:
Simplify the terms inside the brackets:
So, the numerator becomes:
Now, the derivative is:
Finally, cancel one factor of from the numerator and the denominator:
This is the simplified derivative.
Would you like more details, or have any questions?
Related Questions:
- How do you apply the product rule in differentiation?
- Can you explain the chain rule in more detail?
- How does implicit differentiation work in calculus?
- What are some common mistakes when using the quotient rule?
- How does the power rule for differentiation work?
Tip:
When differentiating complex expressions, always simplify as much as possible before applying rules like the quotient rule—it helps avoid unnecessary complications.
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Math Problem Analysis
Mathematical Concepts
Differentiation
Quotient Rule
Product Rule
Chain Rule
Formulas
Quotient rule: d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]^2
Chain rule: d/dx [f(g(x))] = f'(g(x)) * g'(x)
Product rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
Theorems
Quotient rule
Chain rule
Product rule
Suitable Grade Level
Undergraduate Calculus
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