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प्रश्न
Differentiate the following w.r.t. x :
`cos^-1 ((1 - 9^x))/((1 + 9^x)`
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उत्तर
Let y = `cos^-1 ((1 - 9^x))/((1 + 9^x)`
= `cos^-1[(1 - (3^x)^2)/(1 + (3^x)^2)]`
Put 3x = tanθ.
Then θ = tan–1(3x)
∴ y = `cos^-1((1 - tan^2θ)/(1 + tan^2θ))`
= cos–1(cos2θ)
= 2θ
= 2tan–1(3x)
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"[2tan^-1(3^x)]`
= `2"d"/"dx"[tan^-1 (3^x)]`
= `2 xx (1)/(1 + (3^x)^2)."d"/"dx"(3^x)`
= `(2)/(1 + 3^(2x)) xx 3^xlog3`
= `(2.3^xlog3)/(1 + 3^(2x)`
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