Given-that-y-cosx-sinx-find-dy-dx-and-lim-x-0-y-

Question Number 64872 by Rio Michael last updated on 22/Jul/19

G i v e n t h a t

y = {({c o s x}^{})}^{s i n x} f i n d \frac{d y}{d x}

a n d

\underset{x \to 0}{l i m} y

Commented by kaivan.ahmadi last updated on 22/Jul/19

l n y = s i n x l n (c o s x) \Rightarrow \frac{y^{'}}{y} = c o s x l n (c o s x) - \frac{s i n x}{c o s x} s i n x \Rightarrow

\frac{d y}{d x} = {(c o s x)}^{s i n x} (c o s x l n (c o s x) - \frac{{s i n}^{2} x}{c o s x})

{l i m}_{x \to 0} {c o s x}^{s i n x} = 1^{0} = 1

Commented by mathmax by abdo last updated on 22/Jul/19

w e h a v e y = e^{s i n x l n (c o s x)} \Rightarrow \frac{d y}{d x} = \frac{d}{d x} (s i n x l n (c o s x)) e^{s i n x l n (c o s x)}

= {c o s x l n (c o s x) + s i n x \frac{- s i n x}{c o s x}} y (x) \Rightarrow

y^{^{'}} (x) = \frac{{c o s}^{2} x l n (c o s x) - {s i n}^{2} x}{c o s x} \times {(c o s x)}^{s i n x}

Commented by mathmax by abdo last updated on 22/Jul/19

w e h a v e y (x) = e^{s i n x l n (c o s x)} a n d {l i m}_{x \to 0} s i n x l n (c o s x) = 0 \Rightarrow

{l i m}_{x \to 0} y (x) = 1

Commented by Masumsiddiqui399@gmail.com last updated on 23/Jul/19