Question

Find \(F^{\prime}(x)\). $$ F(x)=\int_{1}^{3 x} \frac{1}{t} d t $$

Short answer

The derivative of the given function is \( F^{\prime}(x) = 1/x \).

Step-by-step solution

Apply the Fundamental Theorem of Calculus

Applying the Fundamental Theorem of Calculus, we rewrite the integral function as follows: \( F(x) = ln|3x| - ln|1| \), which simplifies to \( F(x) = ln|3x| \).

Derive the Resulting Function

Now, we need to find the derivative of \( F(x) \), or \( F^{\prime}(x) \). Using the chain rule, which says if we have a composition of functions (say, \( f(g(x)) \)), the derivative of that would be \( f^{\prime}(g(x)) \cdot g^{\prime}(x) \), we obtain \( F^{\prime}(x) = (1/(3x)) \cdot 3 = 1/x \).