Differentiate xsin(x)
Understand the Problem
The question is asking for the differentiation of the function xsin(x). We need to apply the product rule of differentiation, which states that the derivative of a product of two functions is given by f'(x)g(x) + f(x)g'(x).
Answer
\sin(x) + x \cos(x)
Answer for screen readers
The final answer is \sin(x) + x \cos(x)
Steps to Solve
- Identify the functions to differentiate
We have the function $f(x) = x \sin(x)$. Let $f(x) = x$ and $g(x) = \sin(x)$. We need to apply the product rule of differentiation.
- Apply the product rule
The product rule states that $(fg)' = f'g + fg'$. Here, $f(x) = x$ and $g(x) = \sin(x)$.
$$ (x \sin(x))' = (x)' \cdot \sin(x) + x \cdot (\sin(x))' $$
- Find the derivatives of $f(x)$ and $g(x)$
The derivative of $f(x) = x$ is $f'(x) = 1$.
The derivative of $g(x) = \sin(x)$ is $g'(x) = \cos(x)$.
- Substitute the derivatives into the product rule formula
Substitute the derivatives we found into the product rule formula:
$$ (x \sin(x))' = 1 \cdot \sin(x) + x \cdot \cos(x) $$
- Simplify the expression
Combine the terms to get the final derivative:
$$ x \sin(x)' = \sin(x) + x \cos(x) $$
The final answer is \sin(x) + x \cos(x)
More Information
The differentiation of the function $x\sin(x)$ reveals how the rate of change of the function behaves, showing the sum of the sine function and the product of $x$ and cosine function.
Tips
A common mistake when applying the product rule is forgetting to differentiate both functions involved. Make sure to find the derivative of each function before applying the rule.
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