# Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) =...

## Question:

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

{eq}h(x) = \int_{1}^{e^x} \, 4 \ln(t) \, \mathrm{d}t {/eq}

## Definite Integrals:

The fundamental theorem of calculus for finding the differentiation of any expression can be useful. The rule that it follows is derivative from the concept of limits. The formula is: {eq}\frac{d\left ( \int_{a}^{g(x)} f(t) dt\right )}{dx}=\frac{d(g(x))}{dx} f(g(x)) \\ {/eq}

The given integral expression is:

{eq}h(x) = \int_{1}^{e^x} \, 4 \ln(t) \, \mathrm{d}t {/eq}

Now we are finding the derivative as follows:

{eq}\frac{d(h(x))}{dx}= \frac{d\left ( \int_{1}^{e^x} \, 4 \ln(t) \, \mathrm{d}t \right )}{dx}\\ {/eq}

Now using the fundamental theorem that is as per the formula below:

{eq}\frac{d\left ( \int_{a}^{g(x)} f(t) dt\right )}{dx}=\frac{d(g(x))}{dx} f(g(x)) \\ {/eq}

So the given expression will be now;

{eq}\frac{d(h(x))}{dx}= \frac{d\left ( \int_{1}^{e^x} \, 4 \ln(t) \, \mathrm{d}t \right )}{dx}\\ =\frac{d(e^x)}{dx} 4 \ln(e^x) \\ =4 e^x \ln(e^x)\\ \Rightarrow h'(x)= 4 e^x \ln(e^x)\\ {/eq}

So this is the required derivative of the given integral expression.