What is the derivative of (e^{x^3}/(sin(2x))) ?

The derivative of (e^{x^3}/(sin(2x))) is 3e^{x^3}x^2csc(2x)-2e^{x^3}cot(2x)csc(2x)

What is the first derivative of (e^{x^3}/(sin(2x))) ?

The first derivative of (e^{x^3}/(sin(2x))) is 3e^{x^3}x^2csc(2x)-2e^{x^3}cot(2x)csc(2x)

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\frac{d}{d x} (\frac{e ^{x^{3}}}{sin ( 2 x )})

3 e^{x^{3}} x^{2} csc (2 x) - 2 e^{x^{3}} cot (2 x) csc (2 x)

Solution steps

\frac{d}{d x} (\frac{e ^{x^{3}}}{sin ( 2 x )})

Simplify

\frac{e ^{x^{3}}}{sin ( 2 x )} : e^{x^{3}} csc (2 x)

= \frac{d}{d x} (e^{x^{3}} csc (2 x))

Apply the Product Rule:

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

f = e^{x^{3}}, g = csc (2 x)

= \frac{d}{d x} (e^{x^{3}}) csc (2 x) + \frac{d}{d x} (csc (2 x)) e^{x^{3}}

\frac{d}{d x} (e^{x^{3}}) = e^{x^{3}} \cdot 3 x^{2}

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

= e^{x^{3}} \cdot 3 x^{2} csc (2 x) + (- cot (2 x) csc (2 x) \cdot 2) e^{x^{3}}

Simplify

= 3 e^{x^{3}} x^{2} csc (2 x) - 2 e^{x^{3}} cot (2 x) csc (2 x)

What is the derivative of (e^{x^3}/(sin(2x))) ?
The derivative of (e^{x^3}/(sin(2x))) is 3e^{x^3}x^2csc(2x)-2e^{x^3}cot(2x)csc(2x)
What is the first derivative of (e^{x^3}/(sin(2x))) ?
The first derivative of (e^{x^3}/(sin(2x))) is 3e^{x^3}x^2csc(2x)-2e^{x^3}cot(2x)csc(2x)