derivative xsinx
Understand the Problem
The question is asking for the derivative of the function xsin(x). To solve this, we will use the product rule of differentiation, which states that if you have a product of two functions, the derivative is the first function times the derivative of the second plus the second function times the derivative of the first.
Answer
The derivative is given by $f'(x) = \sin(x) + x \cos(x)$.
Answer for screen readers
The derivative of the function $f(x) = x \sin(x)$ is given by:
$$ f'(x) = \sin(x) + x \cos(x) $$
Steps to Solve
- Identify the functions for the product rule
In the function $f(x) = x \sin(x)$, we identify the two functions:
Let $u = x$ and $v = \sin(x)$.
- Differentiate both functions
Now we will find the derivatives of both functions:
$$ u' = \frac{d}{dx}(x) = 1 $$
$$ v' = \frac{d}{dx}(\sin(x)) = \cos(x) $$
- Apply the product rule
Using the product rule, we find the derivative of the function:
$$ f'(x) = u'v + uv' $$
This becomes:
$$ f'(x) = (1)(\sin(x)) + (x)(\cos(x)) $$
- Combine the terms
Now, we can put it all together into a single expression:
$$ f'(x) = \sin(x) + x \cos(x) $$
The derivative of the function $f(x) = x \sin(x)$ is given by:
$$ f'(x) = \sin(x) + x \cos(x) $$
More Information
The product rule is a fundamental concept in calculus that allows us to differentiate products of functions. Understanding this rule is essential for tackling more complex functions in calculus.
Tips
- Forgetting to apply the product rule correctly, which can lead to incorrect derivatives.
- Miscalculating the derivatives of $u$ or $v$. Double-check each derivative to avoid errors.
