What is the derivative of 6x^3sqrt(x)csc(x) ?

The derivative of 6x^3sqrt(x)csc(x) is 6((7x^{5/2}csc(x))/2-x^{7/2}cot(x)csc(x))

What is the first derivative of 6x^3sqrt(x)csc(x) ?

The first derivative of 6x^3sqrt(x)csc(x) is 6((7x^{5/2}csc(x))/2-x^{7/2}cot(x)csc(x))

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\frac{d}{d x} (6 x^{3} x csc (x))

6 (\frac{7 x ^{\frac{5}{2}} csc ( x )}{2} - x^{\frac{7}{2}} cot (x) csc (x))

Solution steps

\frac{d}{d x} (6 x^{3} x csc (x))

Simplify

6 x^{3} x csc (x) : 6 x^{\frac{7}{2}} csc (x)

= \frac{d}{d x} (6 x^{\frac{7}{2}} csc (x))

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d}{d x} (x^{\frac{7}{2}} csc (x))

Apply the Product Rule:

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

f = x^{\frac{7}{2}}, g = csc (x)

= 6 (\frac{d}{d x} (x^{\frac{7}{2}}) csc (x) + \frac{d}{d x} (csc (x)) x^{\frac{7}{2}})

\frac{d}{d x} (x^{\frac{7}{2}}) = \frac{7 x ^{\frac{5}{2}}}{2}

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

= 6 (\frac{7 x ^{\frac{5}{2}}}{2} csc (x) + (- cot (x) csc (x)) x^{\frac{7}{2}})

Simplify

6 (\frac{7 x ^{\frac{5}{2}}}{2} csc (x) + (- cot (x) csc (x)) x^{\frac{7}{2}}) : 6 (\frac{7 x ^{\frac{5}{2}} csc ( x )}{2} - x^{\frac{7}{2}} cot (x) csc (x))

= 6 (\frac{7 x ^{\frac{5}{2}} csc ( x )}{2} - x^{\frac{7}{2}} cot (x) csc (x))

a re a x^{2}, x + 2, - 1, 2

\int sin^{5} (2 t) cos (2 t) d t

a re a 3 x^{3} - 2 x, - 3, 3

\frac{\partial}{\partial y} (\frac{1}{t ( y ^{2} - 9 )})

a re a f (x) = \frac{x}{9 - x ^{2}}, g (y) = 0, y = 2

What is the derivative of 6x^3sqrt(x)csc(x) ?
The derivative of 6x^3sqrt(x)csc(x) is 6((7x^{5/2}csc(x))/2-x^{7/2}cot(x)csc(x))
What is the first derivative of 6x^3sqrt(x)csc(x) ?
The first derivative of 6x^3sqrt(x)csc(x) is 6((7x^{5/2}csc(x))/2-x^{7/2}cot(x)csc(x))