Math Problem Statement
Solution
The problem asks to compute the derivative of the expression with respect to .
Solution:
We are tasked to find:
Step 1: Use the Quotient Rule
The derivative of a fraction is given by the quotient rule:
Here:
Step 2: Compute and
-
, so:
-
, so:
Step 3: Substitute into the Quotient Rule
[ \frac{d}{dx} \left( \frac{\sin(x^2)}{x} \right) = \frac{\left[\cos(x^2) \cdot 2x \cdot x\right] - \left[\sin(x^2) \cdot 1\right]}{x^2}. ]
Simplify the numerator: so the numerator becomes:
Thus:
Step 4: Simplify the Expression
Divide each term in the numerator by :
Final Answer:
Do you want further clarification or a step-by-step explanation for any part of the process?
Related Questions:
- How does the chain rule work in differentiation?
- Can you explain the quotient rule in simpler terms with examples?
- What is the derivative of without dividing by ?
- How can we use trigonometric derivatives in practical applications?
- What happens to this derivative as ?
Tip:
When working with derivatives, always identify whether the product, chain, or quotient rule applies before jumping into calculations!
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Quotient 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)
Theorems
Quotient Rule
Chain Rule
Suitable Grade Level
College-level Calculus or Advanced High School Calculus (Grades 11-12)