Question

Differentiate :

$\\ \bf \: y = {(sin \: x)}^{ log \: x }$
​

Answer 1

Answer:

\large \clubs \: \bf Given : -♣ Given: −

The sum of ages of father and daughter is 52 years.

Six years ago, the product of their ages was 175.

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\large \clubs \: \bf To \: Find : -♣ ToFind: −

Their Present Ages

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\large \clubs \: \bf Solution : -♣ Solution: −

Let,

Present Age of Father = x years

Present Age of Daughter = y years

✏ According To Question :

\begin{gathered} \text{Sum of Their Ages = 52 years} \\ \end{gathered}

Sum of Their Ages = 52 years

\begin{gathered}:\longmapsto \text{x + y = 52} \\ \end{gathered}

:⟼x + y = 52

\begin{gathered}:\longmapsto \bf x = 52 - y \: - - - (1) \\ \end{gathered}

:⟼x=52−y−−−(1)

《 6 Years Ago 》

Age of Father = (x - 6) years

Age of Daughter =(y - 6) years

✏ According To Question :

\begin{gathered} \red{ (\text x - 6)(\text y - 6) = 175} \\ \end{gathered}

(x−6)(y−6)=175

\begin{gathered}:\longmapsto\text x\text y - 6\text x - 6\text y + 36 = 175 \\ \end{gathered}

:⟼xy−6x−6y+36=175

\begin{gathered}:\longmapsto\text x\text y - 6\text x - 6\text y = 175 - 36 \\ \end{gathered}

:⟼xy−6x−6y=175−36

\begin{gathered}:\longmapsto \bf xy - 6x - 6y = 139 \: - - - (2) \\ \end{gathered}

:⟼xy−6x−6y=139−−−(2)

✏ Putting (1) in (2) :

\begin{gathered}:\longmapsto(52 -\text y)\text y - 6(52 - \text y) - 6\text y = 139 \\ \end{gathered}

:⟼(52−y)y−6(52−y)−6y=139

\begin{gathered}:\longmapsto52\text y - \text y {}^{2} - 312 + \cancel{6\text y} - \cancel{6\text y} = 139 \\ \end{gathered}

:⟼52y−y

2

−312+

6y

−

6y

=139

\begin{gathered}:\longmapsto52\text y - \text y {}^{2} = 139 + 312 \\ \end{gathered}

:⟼52y−y

2

=139+312

\begin{gathered} \bf \red{:\longmapsto{{y}^{2} - 52y + 451 = 0 }} \\ \end{gathered}

:⟼y

2

−52y+451=0

\begin{gathered}:\longmapsto\text{y}^{2} - 11\text y - 41\text y + 451 = 0 \\ \end{gathered}

:⟼y

2

−11y−41y+451=0

\begin{gathered}:\longmapsto\text y(\text y - 11) - 41(\text y - 11) = 0 \\ \end{gathered}

:⟼y(y−11)−41(y−11)=0

\begin{gathered}:\longmapsto(\text y - 11)(\text y - 41) = 0 \\ \end{gathered}

:⟼(y−11)(y−41)=0

\begin{gathered}\purple{ \large :\longmapsto \underline {\boxed{{\bf y = 11 \: or \: y = 41} }}} \\ \end{gathered}

:⟼

y=11ory=41

☆ When y = 11 :-

\text x = 52 - 11 \: \: \: \: \bf \{using \: (1) \}x=52−11{using(1)}

\begin{gathered}\purple{ \Large :\longmapsto \underline {\boxed{{\bf x = 41} }}} \\ \end{gathered}

:⟼

x=41

☆ When y = 41 :-

\begin{gathered}\text x = 52 - 41 \\ \end{gathered}

x=52−41

\purple{ \Large :\longmapsto \underline {\boxed{{\bf x = 11} }}}:⟼

x=11

As x is Age father and y is age of daughter

\LARGE \orange{\bf\therefore\:\: x > y}∴x>y

Hence,

\purple{ \large \underline {\boxed{{\bf x = 41 \: \: and \: \: y =11 } }}}

x=41andy=11

Therefore,

\begin{gathered} \pink{\begin{cases} \bf Age \: of \: Father = 41 \: years \\ \\ \bf Age \: of \: Daughter = 11years \end{cases}}\end{gathered}

⎩

⎪

⎪

⎨

⎪

⎪

⎧

AgeofFather=41years

AgeofDaughter=11years

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Answer 2

Question :

$\\ \bf \: y = {(sin \: x)}^{ log \: x }$

• calculate dy/dx.

Solution :

$\underline \bold \purple{Refer \: To \: The \: Attachment}$