Math Problem Statement

Derivative of y=xcoth(1+x^2)

Solution

To find the derivative of the function y=xcoth(1+x2)y = x \coth(1 + x^2), we will apply the product rule and the chain rule.

Given: y=xcoth(1+x2)y = x \coth(1 + x^2)

Step 1: Apply the Product Rule

The product rule states: ddx(uv)=uv+uv\frac{d}{dx}(u \cdot v) = u' \cdot v + u \cdot v' where u=xu = x and v=coth(1+x2)v = \coth(1 + x^2).

Step 2: Differentiate Each Component

  1. Differentiate u=xu = x: u=ddx(x)=1u' = \frac{d}{dx}(x) = 1

  2. Differentiate v=coth(1+x2)v = \coth(1 + x^2): v=ddx(coth(1+x2))v' = \frac{d}{dx}(\coth(1 + x^2)) To differentiate coth(1+x2)\coth(1 + x^2), we use the chain rule. The derivative of coth(u)\coth(u) is \csch2(u)-\csch^2(u), so: v=\csch2(1+x2)ddx(1+x2)v' = -\csch^2(1 + x^2) \cdot \frac{d}{dx}(1 + x^2) The derivative of 1+x21 + x^2 with respect to xx is: ddx(1+x2)=2x\frac{d}{dx}(1 + x^2) = 2x Therefore: v=\csch2(1+x2)2xv' = -\csch^2(1 + x^2) \cdot 2x

Step 3: Combine Using the Product Rule

Now, substitute back into the product rule: dydx=1coth(1+x2)+x(2x\csch2(1+x2))\frac{dy}{dx} = 1 \cdot \coth(1 + x^2) + x \cdot \left(-2x \csch^2(1 + x^2)\right) Simplifying, we get: dydx=coth(1+x2)2x2\csch2(1+x2)\frac{dy}{dx} = \coth(1 + x^2) - 2x^2 \csch^2(1 + x^2)

So, the derivative is: dydx=coth(1+x2)2x2\csch2(1+x2)\boxed{\frac{dy}{dx} = \coth(1 + x^2) - 2x^2 \csch^2(1 + x^2)}

Would you like more details or have any questions?

Here are five related questions to consider:

  1. What is the derivative of coth(x)\coth(x) with respect to xx?
  2. How would the derivative change if the function was y=xcoth(x2)y = x \coth(x^2) instead?
  3. Can you find the second derivative of the function y=xcoth(1+x2)y = x \coth(1 + x^2)?
  4. What are the properties of the hyperbolic cotangent function coth(x)\coth(x)?
  5. How would you differentiate y=coth(1+x2)y = \coth(1 + x^2) without the xx factor?

Tip: Understanding the chain rule is crucial when dealing with composite functions, especially with hyperbolic functions.

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Math Problem Analysis

Mathematical Concepts

Differentiation
Hyperbolic Functions
Chain Rule
Product Rule

Formulas

Product rule
Chain rule

Theorems

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Suitable Grade Level

Undergraduate