derivative of 1/(sin(x)) — step-by-step solution

To simplify differentiation, we rewrite

\frac{1}{\sin (x)}

{\sin (x)}^{-1}

Let's differentiate:

\frac{d}{d x} ({\sin (x)}^{-1})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

\sin (x)

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({\sin (x)}^{-1})

-1 \cdot \frac{d}{d x} (\sin (x)) \cdot {\sin (x)}^{-2}

-1 \cdot \frac{d}{d x} (\sin (x)) \cdot {\sin (x)}^{-2}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot \frac{d}{d x} (\sin (x)) \cdot {\sin (x)}^{-2}

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

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

ⓘ The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

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

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

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

ⓘ The derivative of sin(x) is cos(x):

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

\cos (x)

▶Derivative:

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