What is the derivative of (1+sin(x)/(x+cos(x))) ?

The derivative of (1+sin(x)/(x+cos(x))) is (xcos(x))/((cos(x)+x)^2)

What is the first derivative of (1+sin(x)/(x+cos(x))) ?

The first derivative of (1+sin(x)/(x+cos(x))) is (xcos(x))/((cos(x)+x)^2)

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\frac{d}{d x} (\frac{1 + sin ( x )}{x + cos ( x )})

\frac{x cos ( x )}{( cos ( x ) + x ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{1 + sin ( x )}{x + cos ( x )})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 1 + sin ( x ) ) ( x + cos ( x ) ) - \frac{d}{d x} ( x + cos ( x ) ) ( 1 + sin ( x ) )}{( x + cos ( x ) ) ^{2}}

\frac{d}{d x} (1 + sin (x)) = cos (x)

\frac{d}{d x} (x + cos (x)) = 1 - sin (x)

= \frac{cos ( x ) ( x + cos ( x ) ) - ( 1 - sin ( x ) ) ( 1 + sin ( x ) )}{( x + cos ( x ) ) ^{2}}

Rewrite using trig identities

= \frac{x cos ( x )}{( cos ( x ) + x ) ^{2}}

What is the derivative of (1+sin(x)/(x+cos(x))) ?
The derivative of (1+sin(x)/(x+cos(x))) is (xcos(x))/((cos(x)+x)^2)
What is the first derivative of (1+sin(x)/(x+cos(x))) ?
The first derivative of (1+sin(x)/(x+cos(x))) is (xcos(x))/((cos(x)+x)^2)