1. ## Simple trig differentiation

If f(x) = 8x arcsin x, find f'(x).

I know that we have to apply the product rule, but something isn't adding up (no pun intended).

My procedure;
f'(x) = 8 (arcsin x) + 8x * 1 / (1+x^2)^2
f'(x) = 8 arcsin x + 8x / (1+8x^2)^1/2

What went wrong?

2. Originally Posted by Archduke01
If f(x) = 8x arcsin x, find f'(x).

I know that we have to apply the product rule, but something isn't adding up (no pun intended).

My procedure;
f'(x) = 8 (arcsin x) + 8x * 1 / (1+x^2)^2
f'(x) = 8 arcsin x + 8x / (1+8x^2)^1/2

What went wrong?
$\int\frac{dx}{\sqrt{1+x^2}}=\text{arcsinh}(x)$...hint hint.

3. We know the derivative of arcsin (x)=1/sqrt(1-x^2).

For the product rule, we want the FIRST times the derivative of the second PLUS the SECOND times the derivative of the first.

Here, the first is 8x and the second is arcsin (x).

I get f'(x)=(8x/sqrt(1-x^2))+8 arcsin (x)