Question

# Use Part 2 of the fundamental Theorem of Calculus to find the derivatives. \frac{d}{dx}\int_{1}^{x}\ln t dt

Derivatives
Use Part 2 of the fundamental Theorem of Calculus to find the derivatives.
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}$$

2021-05-03
Step 1
Given
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}$$
Step 2
Solution
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\int_{{{1}}}^{{{x}}}}{\ln{{t}}}{\left.{d}{t}\right.}$$
$$\displaystyle={\ln{{x}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}-{\ln{{1}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{1}$$
$$\displaystyle={\ln{{x}}}-{0}$$
$$\displaystyle={\ln{{x}}}$$