What is the derivative of (arcsin(x)/(sin(x))) ?

The derivative of (arcsin(x)/(sin(x))) is (csc(x))/(sqrt(1-x^2))-cot(x)csc(x)arcsin(x)

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

The first derivative of (arcsin(x)/(sin(x))) is (csc(x))/(sqrt(1-x^2))-cot(x)csc(x)arcsin(x)

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\frac{d}{d x} (\frac{arcsin ( x )}{sin ( x )})

\frac{csc ( x )}{1 - x ^{2}} - cot (x) csc (x) arcsin (x)

Solution steps

\frac{d}{d x} (\frac{arcsin ( x )}{sin ( x )})

Simplify

\frac{arcsin ( x )}{sin ( x )} : arcsin (x) csc (x)

= \frac{d}{d x} (arcsin (x) csc (x))

Apply the Product Rule:

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

f = arcsin (x), g = csc (x)

= \frac{d}{d x} (arcsin (x)) csc (x) + \frac{d}{d x} (csc (x)) arcsin (x)

\frac{d}{d x} (arcsin (x)) = \frac{1}{1 - x ^{2}}

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

= \frac{1}{1 - x ^{2}} csc (x) + (- cot (x) csc (x)) arcsin (x)

Simplify

\frac{1}{1 - x ^{2}} csc (x) + (- cot (x) csc (x)) arcsin (x) : \frac{csc ( x )}{1 - x ^{2}} - cot (x) csc (x) arcsin (x)

= \frac{csc ( x )}{1 - x ^{2}} - cot (x) csc (x) arcsin (x)

x \to 0 lim (\frac{x ^{4}}{x})

x \to \frac{3}{7} lim (\frac{∣ 3 - 7 x ∣}{21 x ^{2} - 9 x})

x \to - 3 - lim (\frac{2 x}{x + 3})

y = sin^{2} (x), y = sin^{3} (x), [0, π]

What is the derivative of (arcsin(x)/(sin(x))) ?
The derivative of (arcsin(x)/(sin(x))) is (csc(x))/(sqrt(1-x^2))-cot(x)csc(x)arcsin(x)
What is the first derivative of (arcsin(x)/(sin(x))) ?
The first derivative of (arcsin(x)/(sin(x))) is (csc(x))/(sqrt(1-x^2))-cot(x)csc(x)arcsin(x)