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

To ease differentiation, the expression

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

is turned into

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

Let's perform the differentiation:

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

We differentiate using:

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

(where f =

1 + \sin (x)

)

ⓘ The derivative of x^n is n*x^(n-1):

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

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

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

ⓘ Multiplying by -1 gives the opposite:

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

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

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

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

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

ⓘ The derivative of a sum is the sum of the derivatives:

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

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

- \frac{0 + \frac{d}{d x} (\sin (x))}{{(1 + \sin (x))}^{2}}

ⓘ Adding 0 changes nothing:

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

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

■The derivative of sin(x) is cos(x):

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

\cos (x)

- \frac{\cos (x)}{{(1 + \sin (x))}^{2}}

▶Derivative:

- \frac{\cos (x)}{{(\sin (x) + 1)}^{2}}