Find the derivative of the following function-fx-cos x-1-sin x

Here f(x)= cos x/1+sin x ∴ f'(x)= d/dx [ cos x/1+sin x ] = (1+sin x)d/dx(cos x) cos x.d/dx(1+sin x)/(1+sin x)^2 = (1+sin x)( sin x) cos x(cos x)/(1+sin x)^2 ...

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Byju's Answer

Standard XI

Mathematics

Left Hand Derivative

Find the deri...

Question

Find the derivative of the following function:
f(x) = $\frac{c o s x}{1 + s i n x}$

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Solution

$H e r e f (x) = \frac{c o s x}{1 + s i n x}$

$∴ f^{'} (x) = \frac{d}{d x} [\frac{c o s x}{1 + s i n x}]$

$= \frac{(1 + s i n x) \frac{d}{d x} (c o s x) - c o s x . \frac{d}{d x} (1 + s i n x)}{(1 + s i n x)^{2}}$

$= \frac{(1 + s i n x) (- s i n x) - c o s x (c o s x)}{(1 + s i n x)^{2}}$

$= \frac{- s i n x - s i n^{2} x - c o s^{2} x}{(1 + s i n x)^{2}}$

$= \frac{- s i n x - (s i n^{2} x + c o s^{2} x)}{(1 + s i n x)^{2}}$

$= \frac{- s i n x - 1}{(1 + s i n x)^{2}} = \frac{- (1 + s i n x)}{(1 + s i n x)^{2}}$

$= \frac{- 1}{1 + s i n x}$

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247

Find the derivative of the following function: f(x) = cos x1+sin x

Heref(x)=cos x1+sin x ∴f′(x)=ddx[cos x1+sin x] =(1+sin x)ddx(cos x)−cos x.ddx(1+sin x)(1+sin x)2 =(1+sin x)(−sin x)−cos x(cos x)(1+sin x)2 =−sin x−sin2x−cos2x(1+sin x)2 =−sin x−(sin2x+cos2x)(1+sin x)2 =−sin x−1(1+sin x)2=−(1+sin x)(1+sin x)2 =−11+sin x

Find the derivative of the following function:
f(x) = $\frac{c o s x}{1 + s i n x}$