Math Problem Statement

Find the derivative of (sin(x^2))/x with respect to x.

Solution

The problem asks to compute the derivative of the expression sin(x2)x\frac{\sin(x^2)}{x} with respect to xx.

Solution:

We are tasked to find: ddx(sin(x2)x)\frac{d}{dx} \left( \frac{\sin(x^2)}{x} \right)

Step 1: Use the Quotient Rule

The derivative of a fraction f(x)g(x)\frac{f(x)}{g(x)} is given by the quotient rule: ddx(f(x)g(x))=f(x)g(x)f(x)g(x)g(x)2.\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}.

Here: f(x)=sin(x2)andg(x)=x.f(x) = \sin(x^2) \quad \text{and} \quad g(x) = x.

Step 2: Compute f(x)f'(x) and g(x)g'(x)

  1. f(x)=sin(x2)f(x) = \sin(x^2), so: f(x)=cos(x2)2x(using the chain rule).f'(x) = \cos(x^2) \cdot 2x \quad \text{(using the chain rule)}.

  2. g(x)=xg(x) = x, so: g(x)=1.g'(x) = 1.

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: cos(x2)2xx=2x2cos(x2),\cos(x^2) \cdot 2x \cdot x = 2x^2 \cos(x^2), so the numerator becomes: 2x2cos(x2)sin(x2).2x^2 \cos(x^2) - \sin(x^2).

Thus: ddx(sin(x2)x)=2x2cos(x2)sin(x2)x2.\frac{d}{dx} \left( \frac{\sin(x^2)}{x} \right) = \frac{2x^2 \cos(x^2) - \sin(x^2)}{x^2}.

Step 4: Simplify the Expression

Divide each term in the numerator by x2x^2: ddx(sin(x2)x)=2cos(x2)sin(x2)x2.\frac{d}{dx} \left( \frac{\sin(x^2)}{x} \right) = 2\cos(x^2) - \frac{\sin(x^2)}{x^2}.

Final Answer:

ddx(sin(x2)x)=2cos(x2)sin(x2)x2.\frac{d}{dx} \left( \frac{\sin(x^2)}{x} \right) = 2\cos(x^2) - \frac{\sin(x^2)}{x^2}.


Do you want further clarification or a step-by-step explanation for any part of the process?


Related Questions:

  1. How does the chain rule work in differentiation?
  2. Can you explain the quotient rule in simpler terms with examples?
  3. What is the derivative of sin(x2)\sin(x^2) without dividing by xx?
  4. How can we use trigonometric derivatives in practical applications?
  5. What happens to this derivative as x0x \to 0?

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)