Derivative of \( \displaystyle \frac{\sin{\left(x \right)}}{x + 1} \)

Problem 2.316 · hard

1. \[ \frac{d}{d x} \frac{\sin{\left(x \right)}}{x + 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = \sin{\left(x \right)} \frac{d}{d x} \frac{1}{x + 1} + \frac{\frac{d}{d x} \sin{\left(x \right)}}{x + 1} \]
   productApply the product rule.✓ Proved
3. \[ = \sin{\left(x \right)} \frac{d}{d x} \frac{1}{x + 1} + \frac{\cos{\left(x \right)}}{x + 1} \]
   trigDifferentiate the sine function.✓ Proved
4. \[ = \frac{\cos{\left(x \right)}}{x + 1} - \frac{\sin{\left(x \right)} \frac{d}{d x} \left(x + 1\right)}{\left(x + 1\right)^{2}} \]
   chainApply the chain rule to the power term.✓ Proved
5. \[ = \frac{\cos{\left(x \right)}}{x + 1} - \frac{\sin{\left(x \right)}}{\left(x + 1\right)^{2}} \]
   derivative algebraDifferentiate the inner function x + 1. Simplify the expression by distributing the terms.✓ Proved
6. \[ = \frac{x \cos{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)}}{\left(x + 1\right)^{2}} \]
   algebraCombine the terms over a common denominator.✓ Proved
7. \[ = \frac{\left(x + 1\right) \cos{\left(x \right)} - \sin{\left(x \right)}}{\left(x + 1\right)^{2}} \]
   simplifySimplify the numerator.✓ Proved