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  • Differentiate the following with respect to <img alt="x : \sin ^{-1}\left(\frac{2^{x+1} \cdot 5^x}{1+(100)^x}\right)" src="https://entrancecorner.oncodecogs.com/gif.latex?x%20%3A%20%5Csin

Differentiate the following with respect to

x : \sin ^{-1}\left(\frac{2^{x+1} \cdot 5^x}{1+(100)^x}\right)

Option: 1

\frac{2^{x+5} \cdot 5^x \log 10}{1+(100)^x}


Option: 2

\frac{2^{x+1} \cdot 5^x \log 10}{1+(100)^x}


Option: 3

\frac{2^{x+1} \cdot 10^x \log 10}{1+(100)^x}


Option: 4

\frac{2^{x+1} \cdot 5^x \log 5}{1+(100)^x}


Answers (1)

best_answer

Let y=\sin ^{-1}\left(\frac{2^{x+1} \cdot 5^x}{1+(100)^x}\right)

simplify the given term,

\begin{aligned} & =\sin ^{-1}\left(\frac{2.2^x \cdot 5^x}{1+(10)^{2 x}}\right) \\ & =\sin ^{-1}\left(\frac{2.10^x}{1+(10)^{2 x}}\right) \end{aligned}

put 10^x=\tan \theta

so that ,

\theta=\tan ^{-1}\left(10^x\right)

Now, put the value of  10^x=\tan \theta in the obtained simplified term,

\begin{aligned} & \sin ^{-1}\left(\frac{2.10^x}{1+(10)^{2 x}}\right)=\sin ^{-1}\left(\frac{2 \tan \theta}{1+\tan ^2 \theta}\right) \\ & =\sin ^{-1}(\sin 2 \theta) \\ & =2 \theta \end{aligned}

Put the value of \theta \text { from }(1) \text {, }

2 \theta=2 \tan ^{-1}\left(10^x\right)

then take derivatives,

\begin{aligned} & \frac{d y}{d x}=2 \cdot \frac{1}{1+\left(10^x\right)^2} \cdot \frac{d}{d x}\left(10^x\right) \\ & =\frac{2 \cdot 10^x \log 10}{1+10^{2 x}} \\ & =\frac{2^{x+1} \cdot 5^x \log 10}{1+(100)^x} \end{aligned}

 

Posted by

himanshu.meshram

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