The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is ______.

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KollegeApply · Accepted Answer

The derivative of cos–1(2x2 – 1) w.r.t. cos–1x is 2. Explanation: Let y = cos–1(2x2 – 1) and t = cos–1x Differentiating both the functions w.r.t. x `"dy"/"dx" = "d"/"dx" cos^-1 (2x^2 - 1) and "dt"/"dx" = "d"/"dx" cos^-1x $\Rightarrow$ "dy"/"dx" = (-1)/sqrt(1 - (2x^2 - 1)^2) * "d"/"dx" (2x^2 - 1) and "dt"/"dx" = (-1)/sqrt(1 - x^2) = (-1.4x)/sqrt(1 - (4x^4 + 1 - 4x^2) and "dt"/"dx" = (-1)/sqrt(1 - x^2) = (-4x)/sqrt(1 - 4x^4 - 1 + 4x^2) = (-4x)/sqrt(4x^2 - 4x^4) = (-4x)/(2xsqrt(1 - x^2) $\Rightarrow$ "dy"/"dx" = (-2)/sqrt(1 - x^2) Now "dy"/"dx" = ("dy"/"dx")/("dt"/"dx") = ((-2)/sqrt(1 - x^2))/((-1)/sqrt(1 - x^2))` = 2.

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