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

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

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

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

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

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

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (sin^{2} (x)) = sin (2 x)

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

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

Simplify

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

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

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