d-d-d-dx-sinx-sinx-

Question Number 103894 by Study last updated on 18/Jul/20

\frac{d}{d (\frac{d}{d x} s i n x)} \cdot s i n x = ?

Answered by 1549442205 last updated on 18/Jul/20

Denote I = \frac{d}{d \frac{d}{dx} sinx} sinx = \frac{d}{dcosx} sinx .

Putting \cos x = t . Then

we have I = \frac{d}{dt} \sqrt{1 - t^{2}} = \frac{- t}{\sqrt{1 - t^{2}}} = \frac{- cosx}{sinx}

= - \cot x

Answered by MAB last updated on 18/Jul/20

\frac{d}{d (\frac{d}{d x} s i n x)} \cdot s i n x = \frac{d}{d c o s (x)} s i n (x)

= \frac{d s i n (x)}{d x} \cdot \frac{d x}{d c o s (x)}

= c o s (x) \cdot (- \frac{1}{s i n (x)})

= - c o t (x)

Commented by Study last updated on 18/Jul/20

h o w w e c a n c a l c u l a t e \frac{d x}{d c o s (x)} ?

Commented by bobhans last updated on 18/Jul/20

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

Answered by bobhans last updated on 18/Jul/20

\frac{d (\sin x)}{d (\frac{d (\sin x)}{d x})} = \frac{d (\sin x)}{d (\cos x)} = \frac{d (\sin x)}{d x} \times \frac{d x}{d (\cos x)}

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

= - \cot x

Facebook Tweet Pin