Answer
425.1k+ views
Hint: Differentiation of function means to compute the derivative of that function. A derivative is the rate at which output changes with respect to an input. We will suppose the given value as $\left( {y = {x^x}} \right)$.
Complete step by step solution:
$ y = {x^x} $
Taking log both sides, we will get
$ \log y = \log {x^x} $
We know that by the property of logarithm that $ {\text{log(m}}{{\text{)}}^{\text{a}}}{\text{ = a log m}} $
\[ \Rightarrow \log \,y = x{\text{ }}log{\text{ }}x\]
Differentiating this value with respect to x, we have
$
\dfrac{1}{y}\dfrac{d}{{dx}}y = x\dfrac{d}{{dx}}\log x + \log x\dfrac{d}{{dx}}x \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = x \times \dfrac{1}{x} \times \dfrac{d}{{dx}}x + \log x \times 1 \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = x \times \dfrac{1}{x} \times 1 + \log x \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = 1 + \log x \\
$
\[\dfrac{1}{y}\,\,\dfrac{{dy}}{{dx}} = 1 + \log x\,\,\]
$ \dfrac{{dy}}{{dx}} = y\left( {1 + \log x} \right) $
\[\dfrac{{dy}}{{dx}} = {x^x}\left( {1 + \log \,x} \right)\] \[\left( {\because y = {x^x}} \right)\]
Note: In these types of questions students must take care while calculating the differentiation of \[log\,x\]. Usually students forget to calculate the derivative $ \left( {\dfrac{d}{{dx}}} \right) $ the value \[log\,x\] with respect to their function.
Complete step by step solution:
$ y = {x^x} $
Taking log both sides, we will get
$ \log y = \log {x^x} $
We know that by the property of logarithm that $ {\text{log(m}}{{\text{)}}^{\text{a}}}{\text{ = a log m}} $
\[ \Rightarrow \log \,y = x{\text{ }}log{\text{ }}x\]
Differentiating this value with respect to x, we have
$
\dfrac{1}{y}\dfrac{d}{{dx}}y = x\dfrac{d}{{dx}}\log x + \log x\dfrac{d}{{dx}}x \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = x \times \dfrac{1}{x} \times \dfrac{d}{{dx}}x + \log x \times 1 \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = x \times \dfrac{1}{x} \times 1 + \log x \\
\dfrac{1}{y}\dfrac{{dy}}{{dx}} = 1 + \log x \\
$
\[\dfrac{1}{y}\,\,\dfrac{{dy}}{{dx}} = 1 + \log x\,\,\]
$ \dfrac{{dy}}{{dx}} = y\left( {1 + \log x} \right) $
\[\dfrac{{dy}}{{dx}} = {x^x}\left( {1 + \log \,x} \right)\] \[\left( {\because y = {x^x}} \right)\]
Note: In these types of questions students must take care while calculating the differentiation of \[log\,x\]. Usually students forget to calculate the derivative $ \left( {\dfrac{d}{{dx}}} \right) $ the value \[log\,x\] with respect to their function.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)