Derivative Problem

I'm unsure of my work when completing this problem:

$$y=e^{-x^{2}} \int^{x}_{0} e^{t^{2}} dt + c_{1}e^{-x^{2}}$$

I applied the product rule to the left bit.

$$\frac{dy}{dx}=e^{-x^{2}} e^{x^{2}} + e^{-x^{2}}(-2x)\int^{x}_{0} e^{t^{2}} dt + (-2x)c_{1}e^{-x^{2}}$$

I'm fairly certain I did this wrong.

I believe that:
$$\int^{x}_{0} e^{t^{2}} dt$$
Is a constant with respect to x, so you don't need to use the product rule, just treat it like any other constant

gb7nash
Homework Helper
I believe that:
$$\int^{x}_{0} e^{t^{2}} dt$$
Is a constant with respect to x, so you don't need to use the product rule, just treat it like any other constant

No, this is a function with respect to x, so the product rule is valid here.

So I derived it correctly?

Ray Vickson
Homework Helper
Dearly Missed
I'm certain you did it right. Why do you think otherwise?

RGV

I'm certain you did it right. Why do you think otherwise?

RGV

It's from a problem where I need to verify that it is a solution to:

$$y'+2xy=1$$

I was concerned that the integral still containing the "t" variable wouldn't cancel, however upon re-inspection I think it should.