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Differentiate w.r.t. x
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x+\frac{0\left(x+2\sqrt{\sin(x)}\right)}{x}
Express 0\times \frac{x+2\sqrt{\sin(x)}}{x} as a single fraction.
\frac{xx}{x}+\frac{0\left(x+2\sqrt{\sin(x)}\right)}{x}
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{xx+0\left(x+2\sqrt{\sin(x)}\right)}{x}
Since \frac{xx}{x} and \frac{0\left(x+2\sqrt{\sin(x)}\right)}{x} have the same denominator, add them by adding their numerators.
x+\frac{0x}{x}
Use the distributive property to multiply 0 by x+2\sqrt{\sin(x)}.
x+\frac{0}{x}
Anything times zero gives zero.
x+0
Zero divided by any non-zero term gives zero.
x
Anything plus zero gives itself.
\frac{\mathrm{d}}{\mathrm{d}x}(x+\frac{0\left(x+2\sqrt{\sin(x)}\right)}{x})
Express 0\times \frac{x+2\sqrt{\sin(x)}}{x} as a single fraction.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{xx}{x}+\frac{0\left(x+2\sqrt{\sin(x)}\right)}{x})
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x}{x}.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{xx+0\left(x+2\sqrt{\sin(x)}\right)}{x})
Since \frac{xx}{x} and \frac{0\left(x+2\sqrt{\sin(x)}\right)}{x} have the same denominator, add them by adding their numerators.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+0\left(x+2\sqrt{\sin(x)}\right)}{x})
Multiply x and x to get x^{2}.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+0x}{x})
Use the distributive property to multiply 0 by x+2\sqrt{\sin(x)}.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+0}{x})
Anything times zero gives zero.
\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}}{x})
Anything plus zero gives itself.
\frac{\mathrm{d}}{\mathrm{d}x}(x)
Cancel out x in both numerator and denominator.
x^{1-1}
The derivative of ax^{n} is nax^{n-1}.
x^{0}
Subtract 1 from 1.
1
For any term t except 0, t^{0}=1.