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

The derivative of 3csc(xcos^3(x)) is 3(-cos^3(x)cot(x)csc(x)-3cos^2(x))

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

The first derivative of 3csc(xcos^3(x)) is 3(-cos^3(x)cot(x)csc(x)-3cos^2(x))

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

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

Solution steps

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

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

\frac{d}{d x} (cos^{3} (x)) = - 3 cos^{2} (x) sin (x)

= 3 ((- cot (x) csc (x)) cos^{3} (x) + (- 3 cos^{2} (x) sin (x)) csc (x))

Simplify

3 ((- cot (x) csc (x)) cos^{3} (x) + (- 3 cos^{2} (x) sin (x)) csc (x)) : 3 (- cos^{3} (x) cot (x) csc (x) - 3 cos^{2} (x))

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

What is the derivative of 3csc(xcos^3(x)) ?
The derivative of 3csc(xcos^3(x)) is 3(-cos^3(x)cot(x)csc(x)-3cos^2(x))
What is the first derivative of 3csc(xcos^3(x)) ?
The first derivative of 3csc(xcos^3(x)) is 3(-cos^3(x)cot(x)csc(x)-3cos^2(x))