evaluate the following derivative by inverse function and implicit function rules d/dx x arcsin(x)

To find the derivative of \( y = x \arcsin(x) \) using the product rule and the inverse trigonometric function rule, let's proceed step-by-step.

### Step 1: Rewrite the Expression
Let:
\[
y = x \arcsin(x)
\]

### Step 2: Apply the Product Rule
The product rule for differentiation states that if \( y = u \cdot v \), then:
\[
\frac{d}{dx}(uv) = u'v + uv'
\]
In this case, let:
- \( u = x \) and \( v = \arcsin(x) \)

Now differentiate each part:
1. \( u = x \Rightarrow u' = 1 \)
2. \( v = \arcsin(x) \Rightarrow v' = \frac{1}{\sqrt{1 - x^2}} \) (derivative of the inverse sine function)

### Step 3: Substitute and Differentiate
Substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula:
\[
\frac{d}{dx} \left( x \arcsin(x) \right) = (1) \cdot \arcsin(x) + x \cdot \frac{1}{\sqrt{1 - x^2}}
\]

### Step 4: Simplify
The derivative simplifies to:
\[
\frac{d}{dx} \left( x \arcsin(x) \right) = \arcsin(x) + \frac{x}{\sqrt{1 - x^2}}
\]

### Final Answer
\[
\frac{d}{dx} \left( x \arcsin(x) \right) = \arcsin(x) + \frac{x}{\sqrt{1 - x^2}}
\]

Would you like further details on this derivative evaluation or the rules used?

Here are five related questions to deepen your understanding:

1. How would you differentiate \( x \arccos(x) \) using similar rules?
2. What is the derivative of \( x \arctan(x) \) by the product rule?
3. How can implicit differentiation help in finding the derivative of \( \arcsin(x) \)?
4. How do we find higher-order derivatives for functions involving inverse trigonometric terms?
5. Can you apply the same differentiation method to \( x \ln(x) \)?

**Tip:** When differentiating products involving inverse trigonometric functions, make sure to combine the product rule with the inverse function rule to capture all terms accurately.

Learn how to evaluate the derivative of x arcsin(x) using the product rule and the inverse function rule. Step-by-step solution with detailed explanation of the inverse trigonometric function derivative and product rule application. Suitable for students in Grades 11-12.

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