Question

If $F ^ { \prime } ( x ) =$

• A. $\left( 9 x ^ { 2 } - 4 x \right) \log x$
• B. $\left( 4 x - 9 x ^ { 2 } \right) \log x$
• C. $\left( 9 x ^ { 2 } + 4 x \right) \log x$
• D. None of these

Answer 1

Answer: (A)

$\left( 9 x ^ { 2 } - 4 x \right) \log x$

Answer 2

$F ( x ) = \int _ { x ^ { 2 } } ^ { x ^ { 3 } } \log t d t$

Applying Leibnitz’s theorem,

$F ^ { \prime } ( x ) = \log x ^ { 3 } \cdot \frac { d } { d x } x ^ { 3 } - \log x ^ { 2 } \cdot \frac { d } { d x } x ^ { 2 }$

$= 3 \log x \cdot 3 x ^ { 2 } - 2 \log x \cdot 2 x$ $= \left( 9 x ^ { 2 } - 4 x \right) \log x$.