# Thread: Second Fundamental Theorem of Calculus

1. ## Second Fundamental Theorem of Calculus

Is it possible to use the Second Fundamental Theorem of Calculus on $\displaystyle C(x)$ to find $\displaystyle C'(x)$, if

$\displaystyle C(x)=\sum_{j=1}^{n}{\bigg|N_{j}\int_{x}^{x_j}{k(s) ds}\bigg|}$,

where $\displaystyle N_j$ is some constant, and $\displaystyle k(s)$ is integrable over every interval?

2. Originally Posted by Fourier
Is it possible to use the Second Fundamental Theorem of Calculus on $\displaystyle C(x)$ to find $\displaystyle C'(x)$, if

$\displaystyle C(x)=\sum_{j=1}^{n}{\bigg|N_{j}\int_{x}^{x_j}{k(s) ds}\bigg|}$,

where $\displaystyle N_j$ is some constant, and $\displaystyle k(s)$ is integrable over every interval?
Yes

RonL

3. Would $\displaystyle C'(x)$ be

$\displaystyle C'(x)=\sum_{j=1}^{n}{\bigg|N_j k(x_j)\bigg|} \textrm{ ?}$

If this is correct, then it implies that the derivative is always nonnegative and increasing as the sum increases from 1 to n.

4. Can anyone check my work?

Thanks