1. ## Finding antiderivatives

The antiderivative of the following functions are sought:
1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

Thanks for you help

Schdero

2. Originally Posted by Schdero
The antiderivative of the following functions are sought:
1)(x)^1/2. The solution is, in my booklet, 1/2(x)^1/2
But if you use the quotient rule ( f(x)/g(x)' => f'(x)g(x)-f(x)g'(x)/ (g(x))^2)
you get 1/-4x(x)^1/2. Where's the problem? I know, that if you consider 1/(x)^1/2 as x^-1/2 it works out, i just dont see the mistake with applying the quotient rule

2) 2(x^3)^1/2 Here I'm clueless, how do you find the antiderivative at this example?

3) 3sin(x). to me the solution is 3(-cos(x)). In my booklet the solution is 3(cos(x)), but isn't the derivative of cos -> -sinus?

Thanks for you help

Schdero

Antiderivative means integrating the given function. So, the antiderivative of $x^{\frac{1}{2}}$ would be

$\int x^{\frac{1}{2}} dx = \frac{2}{3} \times x^{\frac{3}{2}} + C$

If you take the derivative of $\frac{2}{3} \times x^{\frac{3}{2}} + C$ with respect to x, you will get what you integrated, that is, $x^{\frac{1}{2}}$

Likewise, the derivative of $3sinx = 3 cos(x)$, and the antiderivative of $3 sinx = -3cosx+C$

Your answer to find the antiderivative of 3sinx is correct! If what you have asked for (that is finding the antiderivative) is correct, may be your booklet has serious flaws!

3. Originally Posted by harish21
If you take the derivative of $\frac{2}{3} \times x^{\frac{3}{2}} + C$ with respect to x, you will get what you integrated.
Sorry I dount understand anything of this.
1)Is $\frac{1}{2(x^{0.5})}$ as antidervative of $x^{0.5}$ henceforth wrong?
2)why does the quotient rule appereantly not apply here or made i a mistake?