   Question Answers

# Differentiate ${{\left( \log x \right)}^{x}}$ with respect to $\log x$ and obtain the answer.  Hint: In this question, the function is the power of the $\log x$ function of x. Therefore, in this case, we can use the chain rule by defining suitable variables and then simplify it to obtain the required answer.

In the question, we have to differentiate ${{\left( \log x \right)}^{x}}$ with respect to $\log x$ i.e. we have to find $\dfrac{d{{\left( \log x \right)}^{x}}}{d\left( \log x \right)}$.

Now, we know that if u and v are two functions of x, then

$\dfrac{du}{dv}=\dfrac{\dfrac{du}{dx}}{\dfrac{dv}{dx}}..............(1.1)$

Therefore, taking $u={{\left( \log x \right)}^{x}}$ and $v=\log x$ in equation (1.1), we obtain
$\dfrac{d{{\left( \log x \right)}^{x}}}{d\left( \log x \right)}=\dfrac{\dfrac{d{{\left( \log x \right)}^{x}}}{dx}}{\dfrac{d\left( \log x \right)}{dx}}=\dfrac{\dfrac{du}{dx}}{\dfrac{dv}{dx}}................(1.2)$

We know that for any numbers a and b

$\log \left( {{a}^{b}} \right)=b\log \left( a \right)..............(1.2)$

Now, we have defined $u$ as $u={{\left( \log x \right)}^{x}}$. Taking logarithm on both sides and using equation (1.2), we obtain

$\log u=\log \left( {{\left( \log x \right)}^{x}} \right)=x\log \left( \log x \right).....(1.3)$

Now, the derivative of log function is given by

$\dfrac{d\log x}{dx}=\dfrac{1}{x}..............(1.4)$

The chain rules is stated as

$\dfrac{d\left( f(g(x)) \right)}{dx}=\dfrac{df(g)}{dg}\times \dfrac{dg(x)}{dx}........(1.5)$

And the derivative of the product of two functions is given by

$\dfrac{d\left( f(x)g(x) \right)}{dx}=g(x)\dfrac{df(x)}{dx}+f(x)\dfrac{dg(x)}{dx}...........(1.6)$

Therefore, differentiating both sides of equation (1.3) and using equations (1.4), (1.5) and (1.6), we get

\begin{align} & \log u=x\log \left( \log x \right) \\ & \Rightarrow \dfrac{d\log u}{dx}=\dfrac{d\left( x\log \left( \log x \right) \right)}{dx} \\ & \Rightarrow \dfrac{d\log u}{du}\dfrac{du}{dx}=\log \left( \log x \right)\dfrac{d\left( x \right)}{dx}+x\dfrac{d\left( \log \left( \log x \right) \right)}{dx} \\ & \Rightarrow \dfrac{1}{u}\dfrac{du}{dx}=\log \left( \log x \right)\times 1+x\times \dfrac{d\left( \log \left( \log x \right) \right)}{d\log x}\dfrac{d\log x}{dx} \\ & \Rightarrow \dfrac{1}{u}\dfrac{du}{dx}=\log \left( \log x \right)+x\times \dfrac{1}{\log x}\times \dfrac{1}{x} \\ & \Rightarrow \dfrac{du}{dx}=u\left( \log \left( \log x \right)+\dfrac{1}{\log x} \right)={{\left( \log x \right)}^{x}}\left( \log \left( \log x \right)+\dfrac{1}{\log x} \right).....(1.7) \\ \end{align}

Similarly, in the denominator, we can use equation (1.4) to obtain

$\dfrac{dv}{dx}=\dfrac{d\log x}{dx}=\dfrac{1}{x}..............(1.8)$

Therefore, from equations (1.2), (1.7) and (1.8), we obtain

\begin{align} & \dfrac{d{{\left( \log x \right)}^{x}}}{d\left( \log x \right)}=\dfrac{\dfrac{du}{dx}}{\dfrac{dv}{dx}}=\dfrac{{{\left( \log x \right)}^{x}}\left( \log \left( \log x \right)+\dfrac{1}{\log x} \right)}{\dfrac{1}{x}} \\ & \Rightarrow \dfrac{d{{\left( \log x \right)}^{x}}}{d\left( \log x \right)}=x{{\left( \log x \right)}^{x}}\left( \log \left( \log x \right)+\dfrac{1}{\log x} \right) \\ \end{align}

Which is the required answer to this question.

Note: In this case, we should note that we should simply take $\log x$ as a constant and use the derivative of the xth power of a constant as $\dfrac{d\left( {{a}^{x}} \right)}{dx}={{a}^{x}}\log a$ because in this formula a is a constant whereas in the question $\log x$ is not a constant but is a function of x.
Difference Between Log and Ln  CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry Formula  CBSE Class 10 Maths Chapter 10 - Circles Formula  Electromagnetic Spectrum X-rays  IMO Maths Olympiad Sample Question Paper for Class 6 to 10  CBSE Class 12 Maths Chapter-6 Application of Derivatives Formula  CBSE Class 10 Maths Chapter 2 - Polynomials Formula  CBSE Class 8 Maths Chapter 15 - Introduction to Graphs Formulas  CBSE Class 6 to 12 Maths Formulas - Important Math Formulas  CBSE Class 10 Maths Chapter 14 - Statistics Formula  Important Questions for CBSE Class 10 Maths & Science with Answers  Important Questions for CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry  Important Questions for CBSE Class 10 Maths Chapter 9 - Some Applications of Trigonometry  Important Questions for CBSE Class 10 Maths Chapter 12 - Areas Related to Circles  Important Questions for CBSE Class 10 Maths, Chapter wise Questions with Answers  Important Questions with Answers for CBSE Class 6 to 12 - All Subjects  Important Questions for CBSE Class 10 Maths Chapter 10 - Circles  Important Questions for CBSE Class 10 Science, Chapter wise Questions with Answers  Important Questions for CBSE Class 12 Maths Chapter 6 - Application of Derivatives  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 10 - A Strange Wrestling Match  CBSE Previous Year Question Papers Class 10 Maths with Solutions  CBSE Class 10 Maths Question Paper 2019 with Solutions - Free PDF  CBSE Class 10 Hindi A Question Paper with Solutions  CBSE Class 10 English Communicative Question Paper with Solutions  CBSE Class 10 Hindi B Question Paper with Solutions  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  CBSE Class 10 Science Board Question Paper 2019 with Solutions  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry  RD Sharma Solutions for Class 10 Maths Chapter 12 - Some Applications of Trigonometry  Textbooks Solutions for CBSE & ICSE Board of Class 6 to 12 Maths & Science  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry  NCERT Solutions for Class 10 Maths Chapter 12 Areas Related to Circles  NCERT Solutions for Class 10 Maths Chapter 8 - Introduction to Trigonometry  NCERT Solutions for Class 12 Maths Chapter 6  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry in Hindi  NCERT Solutions for Class 6 English A pact with the Sun Chapter-10  RD Sharma Solutions for Class 10 Maths Chapter 15 - Areas Related to Circles  