Determine the second derivative of h(x) =xsin(x). Give the answer in the simplest form

To determine the second derivative of \( h(x) = x\sin(x) \), we first need to find the first derivative \( h'(x) \) and then the second derivative \( h''(x) \).

### Step 1: Find the First Derivative
Using the product rule, which states \( (uv)' = u'v + uv' \), we can differentiate \( h(x) = x\sin(x) \).

Let \( u = x \) and \( v = \sin(x) \).

Then,
\[ u' = 1 \]
\[ v' = \cos(x) \]

So,
\[ h'(x) = u'v + uv' = 1 \cdot \sin(x) + x \cdot \cos(x) \]
\[ h'(x) = \sin(x) + x\cos(x) \]

### Step 2: Find the Second Derivative
Now, we differentiate \( h'(x) = \sin(x) + x\cos(x) \) to find \( h''(x) \).

Using the product rule again on \( x\cos(x) \),
\[ \left( x\cos(x) \right)' = x' \cos(x) + x \left( \cos(x) \right)' \]
\[ = 1 \cdot \cos(x) + x \left( -\sin(x) \right) \]
\[ = \cos(x) - x\sin(x) \]

So,
\[ h''(x) = \left( \sin(x) + x\cos(x) \right)' \]
\[ = \left( \sin(x) \right)' + \left( x\cos(x) \right)' \]
\[ = \cos(x) + \left( \cos(x) - x\sin(x) \right) \]
\[ = \cos(x) + \cos(x) - x\sin(x) \]
\[ = 2\cos(x) - x\sin(x) \]

Therefore, the second derivative of \( h(x) = x\sin(x) \) is:
\[ h''(x) = 2\cos(x) - x\sin(x) \]

Would you like more details or have any questions?

Here are 5 questions you might ask:
1. How do you apply the product rule for derivatives?
2. Can you explain the chain rule for derivatives?
3. How do you find the derivative of a trigonometric function?
4. What are common mistakes to avoid when differentiating products of functions?
5. Can you show an example of finding a second derivative for a different function?

Tip: Always carefully apply the product and chain rules when differentiating products or compositions of functions.

Learn how to find the second derivative of h(x) = x*sin(x) step-by-step. This problem involves differentiation using the product rule and derivatives of trigonometric functions. Suitable for advanced high school students or anyone interested in calculus problems.

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