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Numerade Educator

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Problem 78 Hard Difficulty

If $ f $ is continuous and $ g $ and $ h $ are differentiable functions, find a formula for
$$ \displaystyle \frac{d}{dx} \int^{h(x)}_{g(x)} f(t) \, dt $$

Answer

$f(h(x)) \cdot h^{\prime}(x)-f(y(x)) \cdot g^{\prime}(x)$

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Video Transcript

We know that if half of t is the anti derivative of lower case off of teeth and by the fundamental theme of calculus, we know half of h of acts menace off G of Axe gives us the integral from G F ax to H of X is off of DDT. Therefore, we know the derivative is gonna be just remember that you're multiplying by G prime of acts and H prime of X. Because remember these air, the chain and rules we have H and jeez, we must supply the chain rule.