What is the derivative of 1/(sqrt(x+sin(x))) ?

The derivative of 1/(sqrt(x+sin(x))) is -(1+cos(x))/(2(x+sin(x))^{3/2)}

What is the first derivative of 1/(sqrt(x+sin(x))) ?

The first derivative of 1/(sqrt(x+sin(x))) is -(1+cos(x))/(2(x+sin(x))^{3/2)}

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

- \frac{1 + cos ( x )}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}}

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

- \frac{1}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (x + sin (x))

= - \frac{1}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}} \frac{d}{d x} (x + sin (x))

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

= - \frac{1}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}} (1 + cos (x))

Simplify

- \frac{1}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}} (1 + cos (x)) : - \frac{1 + cos ( x )}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}}

= - \frac{1 + cos ( x )}{2 ( x + sin ( x ) ) ^{\frac{3}{2}}}

\frac{d y}{d x} = \frac{4 e ^{4 x}}{e ^{y}}

\int 2 e^{- s t} d t

\frac{( s - 1 )}{s ^{2} + 2 s + 2}

x \to \infty lim (3 - f (x))

8 x^{3} - 36 x^{2} - 96

What is the derivative of 1/(sqrt(x+sin(x))) ?
The derivative of 1/(sqrt(x+sin(x))) is -(1+cos(x))/(2(x+sin(x))^{3/2)}
What is the first derivative of 1/(sqrt(x+sin(x))) ?
The first derivative of 1/(sqrt(x+sin(x))) is -(1+cos(x))/(2(x+sin(x))^{3/2)}