What is (\partial)/(\partial x)(x^{-1/2}cos(x)) ?

The answer to (\partial)/(\partial x)(x^{-1/2}cos(x)) is -(cos(x))/(2x^{3/2)}-(sin(x))/(x^{1/2)}

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(\partial)/(\partial x)(x^{-1/2}cos(x))

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

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

Solution steps

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

Apply the Product Rule:

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

f = x^{- \frac{1}{2}}, g = cos (x)

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

\frac{\partial}{\partial x} (x^{- \frac{1}{2}}) = - \frac{1}{2 x ^{\frac{3}{2}}}

\frac{\partial}{\partial x} (cos (x)) = - sin (x)

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

Simplify

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

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

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

\frac{\partial}{\partial x} (e^{e^{x y}})

x ∣ x - 2 ∣

x \to - 2 lim (2 ∣ x + 2 ∣)

\frac{\partial}{\partial x} (ln (y + x))

What is (\partial)/(\partial x)(x^{-1/2}cos(x)) ?
The answer to (\partial)/(\partial x)(x^{-1/2}cos(x)) is -(cos(x))/(2x^{3/2)}-(sin(x))/(x^{1/2)}