Applying Fundamental Theorem of Calculus on multiplication of functions

Question

I gotta differentiate this, and I don't know if my answer is correct

$$ F(x) = \int_0^{f(x)} f(u)g(u) \, du = $$

Should it be

$$F'(x) = f(f(x)) f'(x) g(f(x)) f'(x)$$

And if it is or not correct, could you please explain why?

score 1 · Answer 1 · answered Apr 19 '19 at 12:51

1

I'll assume that $f$ and $g$ are differentiable. Denote $$ G(x) = \int_0^{x} f(u)g(u) \, du $$ Then we have $$ G'(x) = f(x)g(x). $$ Can you now apply a chain rule to find a derivative of $$ F(x) = G(f(x))? $$

answered Apr 19 '19 at 12:51

Virtuoz

3,656

score 1 · Accepted Answer · answered Apr 19 '19 at 12:51

1

By Leibniz rule for differentiating under the integral sign,

$f(f(x))g(f(x))*f'(x)$

https://en.wikipedia.org/wiki/Leibniz_integral_rule

answered Apr 19 '19 at 12:51

fGDu94

3,916

's Sir, where did you learn Leibniz Rule for differentiating? And in Wich book I can learn about it? Thank you – Max Villegas Apr 19 '19 at 12:55
I think it's in Michael Spivak Calculus book – fGDu94 Apr 19 '19 at 12:57
sir, I tried it but I can't see how you did it. – Max Villegas Apr 19 '19 at 13:15
The function were integrating is independent of $x$, so the term with the partial derivative is just $0$.
$b(x) = f(x)$, $a(x) = 0$. This means that $\frac{d}{dx}b(x) = f'(x)$, $\frac{d}{dx}a(x) = 0$.
– fGDu94 Apr 19 '19 at 13:23

Applying Fundamental Theorem of Calculus on multiplication of functions

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