compute the derivative of an integral with respect to X

[berriyumz]

Compute the derivative (with respect to x) of the function:


The answer is (-3x^2 + 1)((x^3 + x)+(x^3 +x))

I don't understand how you get this answer because I don't understand how to take the derivative of an integral. I know its something like the second FTC, which is that the derivative of an integral of a function is that original function. So does this bean you "un-antidifferentiate" the function? I really don't get it. Please help me.

 

[galactus]

Just use \(\displaystyle \frac{d}{dx}\int_{a}^{g(x)}f(t)dt=f(g(x))g'(x)\)

or

\(\displaystyle \frac{d}{dx}\int_{g(x)}^{a}f(t)dt=-f(g(x))g'(x)\)

Actually, the answer is more like \(\displaystyle x(1-x^{2})(3x^{2}-1)(x^{3}-x+1)\).