# 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.

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}