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  • Differentiate each of the following w.r.t. x sin inverse 1 by root x +1

Differentiate each of the following w.r.t. x
\sin ^{-1}\left(\frac{1}{\sqrt{x+1}}\right)

Answers (1)

Given \sin ^{-1}\left(\frac{1}{\sqrt{x+1}}\right)

Let us assume t=\left(\frac{1}{\sqrt{x+1}}\right)

\\So, y=\sin ^{-1} t$ \\Now, differentiate y w.r.t t \\$\frac{d y}{d t}=\frac{1}{\sqrt{1-t^{2}}}$ \\And, differentiate t w.r.t x \\$\frac{d t}{d x}=\frac{-1}{2 ({x+1})^\frac{3}{2}}$ \\Now, using chain we get $\mathrm{dy} / \mathrm{dx}$ \\$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{dt}} \times \frac{\mathrm{dt}}{\mathrm{dx}}$ \\$\frac{d y}{d x}=\frac{1}{\sqrt{1-t^{2}}} \times-\frac{1}{2 (x+1)^\frac{3}{2}}

Substitute the value of t
\\ \frac{d y}{d x}=\frac{1}{\sqrt{1-\left(\frac{1}{\sqrt{x+1}}\right)^{2}}} \times-\frac{1}{2 \sqrt{x+1}} \times \frac{1}{x+1} \\ \frac{d y}{d x}=\frac{1}{\sqrt{1-\frac{1}{x+1}}} \times-\frac{1}{2 \sqrt{x+1}} \times \frac{1}{x+1}

\\ \frac{d y}{d x}=\frac{1}{\sqrt{\frac{x+1-1}{x+1}}} \times-\frac{1}{2 \sqrt{x+1}} \times \frac{1}{x+1} \\ \frac{d y}{d x}=\frac{\sqrt{x+1}}{\sqrt{x}} \times-\frac{1}{2 \sqrt{x+1}} \times \frac{1}{x+1} \\ \frac{d y}{d x}=\frac{-1}{2 \sqrt{x}}\left(\frac{1}{x+1}\right)

Hence, this is the differentiation of \sin ^{-1}\left(\frac{1}{\sqrt{x+1}}\right).

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infoexpert22

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