# Find: Use the fundamental theorem of calculus to compute the derivative of g(x) if g(x) =...

## Question:

Use the fundamental theorem of calculus to compute the derivative of {eq}g(x) {/eq} if

{eq}g(x) = \int_{5}^{x} \sqrt{u^2 + u+ 4}du {/eq}

## Fundamental Theorem of Calculus to Evaluate Derivatives:

Suppose a function {eq}f(x) {/eq} is integrable.

Then, the derivative of {eq}\int_a^x f(t) \, dt {/eq}, where {eq}a {/eq} is any real number, is simply evaluated by applying the Fundamental Theorem Calculus, {eq}\dfrac{d}{dx} \int_a^x f(t) \, dt = f(x) {/eq}.

The derivative of {eq}g(x) = \int_{5}^{x} \sqrt{u^2 + u+ 4} \, du {/eq} is obtained by applying the Fundamental Theorem of Calculus,

{eq}\dfrac{d}{dx} \int_a^{x} f(t) dt = f(x). {/eq}

Then,

{eq}\begin{align*} g'(x) &= \displaystyle \frac{d}{dx} \left( \int_{5}^{x} \sqrt{u^2 + u+ 4} \, du \right) \\ &\\ &= \displaystyle \color{blue}{\sqrt{x^2 + x+ 4} } \quad \text{[Because the lower bound is a constant and upper bound is x, then by applying the Fundamental Theorem of Calculus, the derivate is obtained by changing u with x]} \\ \end{align*} {/eq}