How do you find the derivative of F(x)=arcsin sqrt sinx?

Apr 7, 2017

Use the chain rule to find the derivative, because your function $F \left(x\right)$ can be expressed as $G \left(h \left(j \left(x\right)\right)\right)$ where $G \left(x\right) = \arcsin \left(x\right)$, $h \left(x\right) = {x}^{\frac{1}{2}}$ and $j \left(x\right) = \sin \left(x\right)$

Explanation:

The chain rule states that the derivative of a composite function is equal to the derivative of the individual composite functions with respect to their inner functions multiplied by the inner functions derivative with respect to x.

So F'(x) = G'(h(j(x)) * h'(j(x)) * j'(x)

Since G'(x) = (1/(sqrt(1-x^2))), G'(h(j(x))) = 1/(sqrt(1 - (sqrt(sin(x)))^2)
And $h ' \left(x\right) = \frac{1}{2} \sqrt{x} , h ' \left(j \left(x\right)\right) = \frac{1}{2} \left(\sqrt{\sin \left(x\right)}\right)$
And $j ' \left(x\right) = \cos \left(x\right)$

Thus the derivative becomes $\left(\frac{1}{\sqrt{1 - {\left(\sqrt{\sin \left(x\right)}\right)}^{2}}}\right) \cdot \left(\frac{1}{2 \left(\sqrt{\sin \left(x\right)}\right)}\right) \cdot \left(\cos \left(x\right)\right)$

Simplifying, we get $\frac{\cos \left(x\right)}{2 \cdot \left(\sqrt{1 - \sin \left(x\right)}\right) \cdot \left(\sqrt{\sin \left(x\right)}\right)}$

This answer can be further simplified using trigonometric identities.

Apr 7, 2017

 F'(x) = cosx/(2 sqrtsinx sqrt(1-sinx)

Explanation:

$F \left(x\right) = \arcsin \sqrt{\sin} x$

Take sines of both sides:

$\sin F \left(x\right) = \sqrt{\sin} x$

Implicit differentiation wrt x:

$\cos F \left(x\right) \cdot F ' \left(x\right) = \frac{1}{2} \frac{1}{\sqrt{\sin}} x \cos x$

Re-arrange:

$F ' \left(x\right) = \cos \frac{x}{2 \sqrt{\sin} x} \cdot \frac{1}{\cos F \left(x\right)}$

$= \cos \frac{x}{2 \sqrt{\sin} x} \cdot \frac{1}{\cos \left(\arcsin \sqrt{\sin} x\right)}$

Now draw a right-angled triangle and you will see that:

$\cos \left(\arcsin \theta\right) = \sqrt{1 - {\theta}^{2}}$, or $\arcsin \theta = \arccos \left(\sqrt{1 - {\theta}^{2}}\right)$

implies F'(x) = cosx/(2 sqrtsinx sqrt(1-sinx) #