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

The derivative of (sin(x^2)/(sin^2(x))) is -2csc^2(x)cot(x)sin(x^2)+2xcsc^2(x)cos(x^2)

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

The first derivative of (sin(x^2)/(sin^2(x))) is -2csc^2(x)cot(x)sin(x^2)+2xcsc^2(x)cos(x^2)

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

- 2 csc^{2} (x) cot (x) sin (x^{2}) + 2 x csc^{2} (x) cos (x^{2})

Solution steps

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

Simplify

\frac{sin ( x ^{2} )}{sin ^{2} ( x )} : csc^{2} (x) sin (x^{2})

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

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = csc^{2} (x), g = sin (x^{2})

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

\frac{d}{d x} (csc^{2} (x)) = - 2 csc^{2} (x) cot (x)

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

= (- 2 csc^{2} (x) cot (x)) sin (x^{2}) + cos (x^{2}) \cdot 2 x csc^{2} (x)

Simplify

= - 2 csc^{2} (x) cot (x) sin (x^{2}) + 2 x csc^{2} (x) cos (x^{2})

y = 6 x^{2} e^{- x}

f (x) = 2^{20}

f (x) = 2 - x

\frac{d}{d x} (\frac{( 3 x ^{2} - 5 ) ^{4}}{( 4 x ^{3} + 2 ) ^{5}})

y = - x^{2} + 4 x

What is the derivative of (sin(x^2)/(sin^2(x))) ?
The derivative of (sin(x^2)/(sin^2(x))) is -2csc^2(x)cot(x)sin(x^2)+2xcsc^2(x)cos(x^2)
What is the first derivative of (sin(x^2)/(sin^2(x))) ?
The first derivative of (sin(x^2)/(sin^2(x))) is -2csc^2(x)cot(x)sin(x^2)+2xcsc^2(x)cos(x^2)