Answer
Verified
414.9k+ views
Hint: In this question remember to differentiate each function separately and use rules such as product rule which is given as; $\dfrac{{d\left( {pqr} \right)}}{{dx}} = qr\dfrac{{dp}}{{dx}} + rp\dfrac{{dq}}{{dx}} + qp\dfrac{{dr}}{{dx}}$ and chain rule i.e. $\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right) \times g'\left( x \right)$(here f and g are two different function), using this information will help you to approach the solution.
Complete step-by-step answer:
According to the given information we have function ${x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Let$y = {x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
And let $u = {x^{x\cos x}}$ and $v = \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Now y = u + v
Differentiating both side with respect to x we get
$\dfrac{{dy}}{{dx}} = \dfrac{{du}}{{dx}} + \dfrac{{dv}}{{dx}}$ (equation 1)
\[u = {x^{x\cos x}}\]
$ \Rightarrow $\[\log u = \log ({x^{x\cos x}})\]
$ \Rightarrow $\[\log u = x\cos x\log x\]
Differentiating both sides with respect to x, we obtain
\[\dfrac{{d\left( {\log u} \right)}}{{dx}} = \dfrac{{d\left( {x\cos x\log x} \right)}}{{dx}}\]
After applying the product rule which is given as; $\dfrac{{d\left( {pqr} \right)}}{{dx}} = qr\dfrac{{dp}}{{dx}} + rp\dfrac{{dq}}{{dx}} + qp\dfrac{{dr}}{{dx}}$
$\dfrac{1}{u}\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(x).\cos x.\log x + x\dfrac{d}{{dx}}(\cos x).\log x + x\cos x.\dfrac{d}{{dx}}(\log x)$
We know that $\dfrac{{d\left( x \right)}}{{dx}} = 1$, $\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x$and $\dfrac{{d\left( {\log x} \right)}}{{dx}} = \dfrac{1}{x}$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = u\left[ {1.\cos x.\log x + x.( - \sin x)\log x + x\cos x.\dfrac{1}{x}} \right]$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = {x^{x\cos x}}(\cos x.\log x - x.\sin x.\log x + \cos x)$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = {x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right]$(equation 2)
$v = \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Applying both sides, we get
$\log v = \log \left( {\dfrac{{{x^2} + 1}}{{{x^2} - 1}}} \right)$
Applying the quotient rule in the above equation i.e. $\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$we get
$\log v = \log ({x^2} + 1) - \log ({x^2} - 1)$
Differentiating both sides with respect to x, we obtain
\[\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \dfrac{{d\left( {\log ({x^2} + 1)} \right)}}{{dx}} - \dfrac{{d\left( {\log ({x^2} - 1)} \right)}}{{dx}}\]
By applying the chain rule i.e. $\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right) \times g'\left( x \right)$(here f and g are two different function) in the above equation we get
\[\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \left( {\dfrac{{d\left( {\log ({x^2} + 1)} \right)}}{{dx}}\dfrac{{d\left( {{x^2} + 1} \right)}}{{dx}}} \right) - \left( {\dfrac{{d\left( {\log ({x^2} - 1)} \right)}}{{dx}}\dfrac{{d\left( {{x^2} - 1} \right)}}{{dx}}} \right)\]
We know that $\dfrac{{d\left( {\log x} \right)}}{{dx}} = \dfrac{1}{x}$and $\dfrac{d}{{dx}}\left( {{x^2} \pm 1} \right) = 2x$
Therefore, $\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \dfrac{{2x}}{{{x^2} + 1}} - \dfrac{{2x}}{{{x^2} - 1}}$
$\dfrac{{dv}}{{dx}} = v\left[ {\dfrac{{2x({x^2} - 1) - 2x({x^2} + 1)}}{{({x^2} + 1)({x^2} - 1)}}} \right]$
$\dfrac{{dv}}{{dx}} = \dfrac{{{x^2} + 1}}{{{x^2} - 1}} \times \left[ {\dfrac{{ - 4x}}{{({x^2} + 1)({x^2} - 1)}}} \right]$
$\dfrac{{dv}}{{dx}} = \dfrac{{ - 4x}}{{{{({x^2} - 1)}^2}}}$(equation 3)
Now substituting the values form (2) and (3) in the equation 1 er get
$\dfrac{{dy}}{{dx}} = {x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right] - \dfrac{{4x}}{{{{({x^2} - 1)}^2}}}$
Therefore, after differentiating this function ${x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$with respect to x we get ${x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right] - \dfrac{{4x}}{{{{({x^2} - 1)}^2}}}$.
Note: In the above solution we came across the term “function” which can be explained as relation between the provided inputs and the outputs of the given inputs such that each input is directly related to the one output. The representation of a function is given by supposing if there is a function “f” that belongs from X to Y. Examples of functions are logarithmic functions, bijective functions, trigonometric functions, binary functions, etc.
Complete step-by-step answer:
According to the given information we have function ${x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Let$y = {x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
And let $u = {x^{x\cos x}}$ and $v = \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Now y = u + v
Differentiating both side with respect to x we get
$\dfrac{{dy}}{{dx}} = \dfrac{{du}}{{dx}} + \dfrac{{dv}}{{dx}}$ (equation 1)
\[u = {x^{x\cos x}}\]
$ \Rightarrow $\[\log u = \log ({x^{x\cos x}})\]
$ \Rightarrow $\[\log u = x\cos x\log x\]
Differentiating both sides with respect to x, we obtain
\[\dfrac{{d\left( {\log u} \right)}}{{dx}} = \dfrac{{d\left( {x\cos x\log x} \right)}}{{dx}}\]
After applying the product rule which is given as; $\dfrac{{d\left( {pqr} \right)}}{{dx}} = qr\dfrac{{dp}}{{dx}} + rp\dfrac{{dq}}{{dx}} + qp\dfrac{{dr}}{{dx}}$
$\dfrac{1}{u}\dfrac{{du}}{{dx}} = \dfrac{d}{{dx}}(x).\cos x.\log x + x\dfrac{d}{{dx}}(\cos x).\log x + x\cos x.\dfrac{d}{{dx}}(\log x)$
We know that $\dfrac{{d\left( x \right)}}{{dx}} = 1$, $\dfrac{d}{{dx}}\left( {\cos x} \right) = - \sin x$and $\dfrac{{d\left( {\log x} \right)}}{{dx}} = \dfrac{1}{x}$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = u\left[ {1.\cos x.\log x + x.( - \sin x)\log x + x\cos x.\dfrac{1}{x}} \right]$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = {x^{x\cos x}}(\cos x.\log x - x.\sin x.\log x + \cos x)$
$ \Rightarrow $$\dfrac{{du}}{{dx}} = {x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right]$(equation 2)
$v = \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$
Applying both sides, we get
$\log v = \log \left( {\dfrac{{{x^2} + 1}}{{{x^2} - 1}}} \right)$
Applying the quotient rule in the above equation i.e. $\log \left( {\dfrac{x}{y}} \right) = \log x - \log y$we get
$\log v = \log ({x^2} + 1) - \log ({x^2} - 1)$
Differentiating both sides with respect to x, we obtain
\[\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \dfrac{{d\left( {\log ({x^2} + 1)} \right)}}{{dx}} - \dfrac{{d\left( {\log ({x^2} - 1)} \right)}}{{dx}}\]
By applying the chain rule i.e. $\dfrac{d}{{dx}}\left[ {f\left( {g\left( x \right)} \right)} \right] = f'\left( {g\left( x \right)} \right) \times g'\left( x \right)$(here f and g are two different function) in the above equation we get
\[\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \left( {\dfrac{{d\left( {\log ({x^2} + 1)} \right)}}{{dx}}\dfrac{{d\left( {{x^2} + 1} \right)}}{{dx}}} \right) - \left( {\dfrac{{d\left( {\log ({x^2} - 1)} \right)}}{{dx}}\dfrac{{d\left( {{x^2} - 1} \right)}}{{dx}}} \right)\]
We know that $\dfrac{{d\left( {\log x} \right)}}{{dx}} = \dfrac{1}{x}$and $\dfrac{d}{{dx}}\left( {{x^2} \pm 1} \right) = 2x$
Therefore, $\dfrac{1}{v}\dfrac{{dv}}{{dx}} = \dfrac{{2x}}{{{x^2} + 1}} - \dfrac{{2x}}{{{x^2} - 1}}$
$\dfrac{{dv}}{{dx}} = v\left[ {\dfrac{{2x({x^2} - 1) - 2x({x^2} + 1)}}{{({x^2} + 1)({x^2} - 1)}}} \right]$
$\dfrac{{dv}}{{dx}} = \dfrac{{{x^2} + 1}}{{{x^2} - 1}} \times \left[ {\dfrac{{ - 4x}}{{({x^2} + 1)({x^2} - 1)}}} \right]$
$\dfrac{{dv}}{{dx}} = \dfrac{{ - 4x}}{{{{({x^2} - 1)}^2}}}$(equation 3)
Now substituting the values form (2) and (3) in the equation 1 er get
$\dfrac{{dy}}{{dx}} = {x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right] - \dfrac{{4x}}{{{{({x^2} - 1)}^2}}}$
Therefore, after differentiating this function ${x^{x\cos x}} + \dfrac{{{x^2} + 1}}{{{x^2} - 1}}$with respect to x we get ${x^{x\cos x}}\left[ {\cos x(1 + \log x) - x\sin x\log x} \right] - \dfrac{{4x}}{{{{({x^2} - 1)}^2}}}$.
Note: In the above solution we came across the term “function” which can be explained as relation between the provided inputs and the outputs of the given inputs such that each input is directly related to the one output. The representation of a function is given by supposing if there is a function “f” that belongs from X to Y. Examples of functions are logarithmic functions, bijective functions, trigonometric functions, binary functions, etc.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE