Mathematics

Mathematics

Problem

If f(x)=sin⁡−1(3x−4x3)f(x) = \sin^{-1}(3x - 4x^3) . Then answer the following The value of f′(12)f'\left(\frac{1}{\sqrt{2}}\right) , is a. −3-3 b. 33 c. −32-3\sqrt{2} d. 323\sqrt{2}
Intuitive explanation

The key is recognizing that 3x−4x33x - 4x^3 is the triple-angle formula for sine: sin⁡(3θ)=3sin⁡θ−4sin⁡3θ\sin(3\theta) = 3\sin\theta - 4\sin^3\theta. This means f(x)=sin⁡−1(sin⁡(3θ))f(x) = \sin^{-1}(\sin(3\theta)) where sin⁡θ=x\sin\theta = x, so f(x)=3θ=3sin⁡−1(x)f(x) = 3\theta = 3\sin^{-1}(x) on the appropriate domain. Once you simplify ff this way, finding f′f' becomes straightforward differentiation—no chain rule messiness needed.

Intuitive explanation

The key is recognizing that 3x−4x33x - 4x^3 is the triple-angle formula for sine: sin⁡(3θ)=3sin⁡θ−4sin⁡3θ\sin(3\theta) = 3\sin\theta - 4\sin^3\theta. This means f(x)=sin⁡−1(sin⁡(3θ))f(x) = \sin^{-1}(\sin(3\theta)) where sin⁡θ=x\sin\theta = x, so f(x)=3θ=3sin⁡−1(x)f(x) = 3\theta = 3\sin^{-1}(x) on the appropriate domain. Once you simplify ff this way, finding f′f' becomes straightforward differentiation—no chain rule messiness needed.

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