given-that-y-ln-1-cos-2-x-find-dy-dx-at-the-point-x-3pi-4-and-if-y-ln-x-2-4-find-dy-dx-at-x-1-

Question Number 72343 by Rio Michael last updated on 27/Oct/19

g i v e n t h a t y = l n (1 + {c o s}^{2} x) f i n d \frac{d y}{d x} a t t h e p o i n t x = \frac{3 π}{4}

a n d i f y = l n (x^{2} + 4) f i n d \frac{d y}{d x} a t x = 1

Commented by mathmax by abdo last updated on 27/Oct/19

y (x) = l n (1 + {c o s}^{2} x) \Rightarrow y^{^{'}} (x) = \frac{- 2 s i n x c o s x}{1 + {c o s}^{2} x} \Rightarrow

y^{^{'}} (\frac{3 π}{4}) = \frac{- 2 s i n (\frac{3 π}{4}) c o s (\frac{3 π}{4})}{1 + {c o s}^{2} (\frac{3 π}{4})} = \frac{2 s i n (\frac{π}{4}) c o s (\frac{π}{4})}{1 + {c o s}^{2} (\frac{π}{4})} = \frac{2 . \frac{1}{\sqrt{2}} . \frac{1}{\sqrt{2}}}{1 + {(\frac{1}{\sqrt{2}})}^{2}}

= \frac{1}{\frac{3}{2}} = \frac{2}{3}

y (x) = l n (x^{2} + 4) \Rightarrow y^{^{'}} (x) = \frac{2 x}{x^{2} + 4} \Rightarrow y^{^{'}} (1) = \frac{2}{5}

Commented by Rio Michael last updated on 30/Oct/19

t h a n k y o u s i r

Commented by mathmax by abdo last updated on 31/Oct/19

y o u a r e w e l c o m e .