If the first derivative of the function fx-sincos -11-22 x-1-22 x with respect to x at x-1 is -b-alog 2- where a- b are coprime- then -4 b-2 a-

Let t=2^x =sin(cos^ 1(1 t^21+t^2)) =sin(sin^ 1(2t1+t^2)) =2t1+t^2 =2.2^x1+2^2x Differentiating f'(x)=(1+2^2x)(2.2^xlog 2) 2.2^x.2^2xlog 2.2(1+2^2x)^2 f'(1)=5 ...

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Byju's Answer

Standard XII

Mathematics

Differentiation Using Substitution

If the first ...

Question

If the first derivative of the function $f (x) = sin ({cos}^{- 1} (\frac{1 - 2^{2 x}}{1 + 2^{2 x}}))$ with respect to $x$ at $x = 1$ is $\frac{- b}{a} log 2,$ where $a, b$ are coprime, then $| 4 b - 2 a | =$

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Solution

Let $t = 2^{x}$
$= sin ({cos}^{- 1} (\frac{1 - t^{2}}{1 + t^{2}}))$
$= sin ({sin}^{- 1} (\frac{2 t}{1 + t^{2}}))$

$= \frac{2 t}{1 + t^{2}}$

$= \frac{{2.2}^{x}}{1 + 2^{2 x}}$
Differentiating
$f^{'} (x) = \frac{(1 + 2^{2 x}) ({2.2}^{x} log 2) - {2.2}^{x} {.2}^{2 x} log 2.2}{{(1 + 2^{2 x})}^{2}}$

$f^{'} (1) = \frac{5.2.2 log 2 - 2.2.4.2 log 2}{(5)^{2}}$

$f^{'} (1) = - \frac{b}{a} log 2 = - \frac{12}{25} log 2$

$∴ a = 25 b = 12$

$| 4 b - 2 a | = | 48 - 50 | = | - 2 | = 2$

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If the first derivative of the function f(x)=sin(cos−1(1−22x1+22x)) with respect to x at x=1 is −balog2, where a,b are coprime, then |4b−2a|=

Let t=2x =sin(cos−1(1−t21+t2)) =sin(sin−1(2t1+t2)) =2t1+t2 =2.2x1+22x Differentiating f′(x)=(1+22x)(2.2xlog2)−2.2x.22xlog2.2(1+22x)2 f′(1)=5.2.2log2−2.2.4.2log2(5)2 f′(1)=−balog2=−1225log2 ∴a=25 b=12 |4b−2a|=|48−50|=|−2|=2

If the first derivative of the function $f (x) = sin ({cos}^{- 1} (\frac{1 - 2^{2 x}}{1 + 2^{2 x}}))$ with respect to $x$ at $x = 1$ is $\frac{- b}{a} log 2,$ where $a, b$ are coprime, then $| 4 b - 2 a | =$