Differentiate sec-1(x)

Let us predict that y = sec-1x

By rewriting in terms of secant, we get

sec y = x

On differentiating with respect to x

sec y tan y. y’=1

Dividing both sides by sec y tan y y’= 1 / sec y tan y

Since sec y = x and from the identity we come to know that

[latex]\tan x=\sqrt{\sec ^{2}y – 1}=\sqrt{x^{2}-1}[/latex] [latex]y’=\frac{1}{x\sqrt{x^{2}-1}}[/latex]

Hence

[latex]\frac{\mathrm{d}}{\mathrm{d} x}sec^{-1}x= \frac{1}{x\sqrt{x^{2}-1}}[/latex]

Answer

[latex]\frac{\mathrm{d}}{\mathrm{d} x}sec^{-1}x= \frac{1}{x\sqrt{x^{2}-1}}[/latex]

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