What is the derivative of $2^arcsin(x)$ ?

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Answer 1 · 2016-08-03T16:26:05

$(2^arcsin(x)ln(2))/sqrt(1-x^2)$

Let:

$y=2^arcsin(x)$

Take the natural logarithm of both sides:

$ln(y)=ln(2^arcsin(x))$

Simplify using logarithm rules:

$ln(y)=arcsin(x)*ln(2)$

Differentiate both sides. You should remember that:

The left-hand side will need the chain rule, similar to implicit differentiation.
On the right-hand side, $ln(2)$ is just a constant.
The derivative of $arcsin(x)$ is $1/sqrt(1-x^2)$ .

Differentiating yields:

$1/y*dy/dx=ln(2)/sqrt(1-x^2)$

Now, solving for $dy/dx$ , the derivative, multiply both sides by $y$ .

$dy/dx=(y*ln(2))/sqrt(1-x^2)$

Write $y$ as $2^arcsin(x)$ :

$dy/dx=(2^arcsin(x)ln(2))/sqrt(1-x^2)$

What is the derivative of 2^arcsin(x)2^arcsin(x)?