Deriving arcsin(1-2e^-t): What Chain Rule to Use?

[goomer]

1. Derive arcsin(1 - 2 e ^-t)



2. The derivative of arcsin is 1/√(1-x^2)



3. I tried using the chain rule for 1 - 2 e ^-t, but that didn't work out. What should I take the chain rule of?

 

[Mark44]

1. Derive arcsin(1 - 2 e ^-t)

That should be "Differentiate arcsin(1 - 2e^(-t))"

2. The derivative of arcsin is 1/√(1-x^2)



3. I tried using the chain rule for 1 - 2 e ^-t, but that didn't work out. What should I take the chain rule of?

$$\frac{d}{du}arcsin(u) = \frac{1}{\sqrt{1 - u^2}}$$

so

$$\frac{d}{dx}arcsin(u) = \frac{d}{du}arcsin(u)~\frac{du}{dx}$$

Can you figure out what u is here?

 

[HallsofIvy]

What you have done so far is almost right. The derivative of arcsin(x) is
$$\frac{1}{1- x^2}$$
and, in $arcsin(1- 2e^{-t})$, $x= 1- 2e^{-t}$
So the derivative of $arcsin(1- 2e^{-t})$
is
$$\frac{1}{1- (1- 2e^{-t})^2}$$
times (the chain rule) the derivative of
$$1- 2e^{-t}$$

 

[goomer]

I'm still not quite sure...is it the chain rule of the entire √(1-u^2) portion?
I'm certain the derivative of arcsin is \frac{1}{\sqrt{1 - x^2}}[/tex] however.

 

[goomer]

Sorry, I don't know how to use the PrettyFont
1/√(1-x^2) is the derivative of arcsin, I mean

 

[thegreenlaser]

I'm still not quite sure...is it the chain rule of the entire √(1-u^2) portion?
I'm certain the derivative of arcsin is \frac{1}{\sqrt{1 - x^2}}[/tex] however.

When applying the chain rule, you only differentiate the 'u' term by itself, not the whole square root term. See Mark44's post.

 

[goomer]

So I am right in trying to differentiate (1 - 2 e ^-t) for the chain rule in my original problem?

 

[micromass]

I'm still not quite sure...is it the chain rule of the entire √(1-u^2) portion?
I'm certain the derivative of arcsin is \frac{1}{\sqrt{1 - x^2}}[/tex] however.

You'll need to put [ tex ] in front of your equation to (without spaces). So you should have typed [ tex ]\frac{1}{\sqrt{1 - x^2}}[ /tex ].
That's how you use the PrettyFont here (called LaTeX). Alternatively, you can also use the x² and x₂ buttons above to create su(b/p)scripts.

Sorry, just wanted to mention that