Let y=2^cos^2x ∴ log y=log(2^cos^2x)=cos^2x.log 2 On diffrentiating w.r.t x, we get 1/y.dy/dx=log 2.d/dx(cos x)^2 ⇒1/y.dy/dx=log2.[2cos x].d/dx(cos x) =log 2. ...

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Byju's Answer

Standard XII

Mathematics

Inequalities of Integrals

Find d/d x2co...

Question

$F i n d \frac{d}{d x} (2^{c o s^{2} x})$

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Solution

Let $y = 2^{c o s^{2} x}$
$∴ l o g y = l o g (2^{c o s^{2} x}) = c o s^{2} x . l o g 2$

On diffrentiating w.r.t x, we get
$\frac{1}{y} . \frac{d y}{d x} = l o g 2. \frac{d}{d x} (c o s x)^{2} \Rightarrow \frac{1}{y} . \frac{d y}{d x} = l o g 2. [2 c o s x] . \frac{d}{d x} (c o s x) = l o g 2. 2. c o s x . (- s i n x) = l o g 2. (- s i n 2 x) ∴ \frac{d y}{d x} = - y . (s i n 2 x) l o g 2$

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Find ddx(2cos2x)

Let y=2cos2x ∴ log y=log(2cos2x)=cos2x.log 2 On diffrentiating w.r.t x, we get 1y.dydx=log 2.ddx(cos x)2⇒1y.dydx=log2.[2cos x].ddx(cos x)=log 2. 2.cos x.(−sin x)=log 2. (−sin2 x)∴ dydx=−y.(sin 2x) log 2

$F i n d \frac{d}{d x} (2^{c o s^{2} x})$