Question

If y = cos^-1 (1 - 4^x / 1 + 4^x ) . Find dy/dx​

Answer 1

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Answer:

Here, we need to find ddx(cos−1(2x+11+4x)).

Now,

y=cos−1(2x+11+4x)y=cos−1(2.2x1+(2x)2)

Substitute 2x=tanθin the above equation.

y=cos−1(2tanθ1+tan2θ)

Now, using the property sin2θ=2tanθ1+tan2θ, we get

y=cos−1(sin2θ)y=cos−1(cosπ2−2θ)y=π2−2θ

Next, substituting θ=tan−12x, we get

y=π2−(2)tan−1(2x)

Differentiate both sides with respect to x using the chain rule.

dydx=0−2(ddx(tan−12x)).(ddx(2x))dydx=−2(11+(2x)2).(2x.log2)dydx=−(2x+1.log21+4x)

Therefore, dydx=−(2x+1.log21+4x)