Question

Assertion (A):The derivative of $(\log x{)}^x$ w.r.t $x$ is $\left(\log x{)}^{x-1}\right[1+\log x\log \left(\log x\right)]$
Reason (R): $\frac{d}{dx}\left\{f\right(x{)}^{g\left(x\right)}\}=f(x{)}^{g\left(x\right)}$ $\left(g\right(x\left)\frac{f{}^{\prime }\left(x\right)}{f\left(x\right)}+g{}^{\prime }\left(x\right)\log \left(f\right(x\left)\right)\right)$

• A. Both A and R are true R is correct reason of A
• B. Both A and R are true R is not correct reason of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true R is correct reason of A

Assertion $(\log x{)}^x=e^{x\log \left(\log x\right)}$

$\frac{d(\log x{)}^x}{dx}=\left(\log x{)}^x\right(\log \left(\log \right(x\left)\right)+\frac{x}{\log \left(x\right)}\frac{1}{x})$

$\left(\log \right(x\left){)}^{x-1}\right(\log \left(x\right)\log \left(\log \right(x\left)\right)+1)$

Reason

$\frac{d\left(f\right(x{)}^{g\left(x\right)}\left)\right)}{dx}=\frac{d\left(e^{g\left(x\right)\log f\left(x\right)}\right)}{dx}$

$=\left(f\right(x\left){)}^{g\left(x\right)}\right(g^{\prime }\left(x\right)\log f\left(x\right)+\frac{g\left(x\right)}{f\left(x\right)}{f}^{\prime }\left(x\right))$