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do's question at Yahoo! Answers regarding differentiating a definite integral

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MarkFL

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Feb 24, 2012
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Here is the question:

Find the derivative of the following function?

( ∫ )(from x^(1/2) to 1) ((s^2)/(1+3s^4))ds using the appropriate form of the Fundamental Theorem of Calculus. F'(x) =?
I have posted a link there to this topic so the OP can see my work.
 
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MarkFL

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Feb 24, 2012
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Hello do,

We are given to evaluate:

\(\displaystyle F(x)=\int_{\sqrt{x}}^1\frac{s^2}{1+3s^4}\,ds\)

Let:

\(\displaystyle f(s)=\frac{d}{ds}F(s)=\frac{s^2}{1+3s^4}\)

Using the anti-derivative form of the fundamental theorem of calculus, we may write:

\(\displaystyle f(x)=\frac{d}{dx}F(x)=\frac{d}{dx}\left(F(1)-F\left(\sqrt{x} \right) \right)\)

Using the fact that the derivative of a constant is zero on the first term, and applying the chain rule on the second term, we find:

\(\displaystyle f(x)=0-\frac{\left(\sqrt{x} \right)^2}{1+3\left(\sqrt{x} \right)^4}\frac{d}{dx}\left(\sqrt{x} \right)=-\frac{x}{2\sqrt{x}\left(1+3x^2 \right)}=-\frac{\sqrt{x}}{2\left(1+3x^2 \right)}\)