# Calculate the derivative \frac{\mathrm{d} }{\mathrm{d} t}\int_3^t \sec(9x+2)dx on what interval...

## Question:

Calculate the derivative {eq}\frac{\mathrm{d} }{\mathrm{d} t}\int_3^t \sec(9x+2)dx {/eq} on what interval the derivative is defined?

## Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) part I states the following

{eq}\displaystyle \frac{d}{dt}\left(\int_a^t f(x)\ dx\right) =f(t), a - \text{ constant }. {/eq}

It is important to note that the theorem requires to have a constant at the lower limit of integration and

the upper limit of integration should be the same variable as the variable with respect to which the derivative is taken.

## Answer and Explanation:

To evaluate the derivative {eq}\displaystyle \frac{\mathrm{d} }{\mathrm{d} t}\int_3^t \sec(9x+2)dx, {/eq}

we will use the FTC part I,

{eq}\displaystyle \boxed{\frac{d}{dt}\left(\int_3^t \sec(9x+2)dx\right) =\sec(9t+2)}, \text{ whenever } sec(9t+2)\neq 0. {/eq}

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