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Question
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left(36 \right)^x} \right\}\] with respect to x.
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Solution
We have,
y = `sin^-1[(2^(x + 1) . 3^x)/(1 + 36^x)]`
y = `sin^-1[(2 . 6^x)/(1 + 6^(2x))]`
Put 6x = tan θ
⇒ θ = tan–1(6x)
Now,
y = `sin^-1[(2 tanθ)/(1 + tan^2θ)]`
y = sin–1 [sin 2θ]
y = 2θ
y = 2tan–1(6x)
`dy/dx = 2/(1 + 6^(2x)) xx d/dx (6^x)`
`dy/dx = 2/(1 + 6^(2x)) xx 6^x log 6`
`dy/dx = 2/(1 + (36)^(x)) xx 6^x log 6`
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f(x) = 3x2 + 6x + 8, x ∈ R
