How do you find the derivative of $f(x) = 3 arcsin(x^2)$ ?

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Answer 1 · 2017-02-20T14:06:24

$dy/dx=(6x)/sqrt(1-x^4)$

Let $y=f(x)$

$y=3arcsin(x^2)$

This suggests that $sin(y/3)=x^2$

Taking the derivative of both sides, we get:

$1/3cos(y/3)dy/dx=2x$

Rearranging and cleaning it up, we get:

$dy/dx=(6x)/(cos(y/3))$

We now need to rewrite $cos(y/3)$ in terms of $x$ . We can do this using $sin^2A+cos^2A=1$

$cos^2(y/3)+sin^2(y/3)=1$ where $sin^2(y/3)=x^4$

$cos^2(y/3)=1-x^4$

$cos(y/3)=sqrt(1-x^4)$

$dy/dx=(6x)/sqrt(1-x^4)$

How do you find the derivative of f(x) = 3 arcsin(x^2)f(x) = 3 arcsin(x^2)?