Question

Sum of the digits of a two-digit number is 11. The given number is less than the number obtained by interchanging the digits by 9. Find the number. ​

Answer 1

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▪Given :-

\bf y = [log \{log(logx) \}] {}^{2}y =[log{log(logx)}]

2

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▪To Calculate :-

\bf \large \color{magenta}{dy/dx}dy/dx

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▪Formulae Used :-

\begin{gathered} \bigstar \: \bf \frac{d}{dx} f(x) {}^{2} = 2f(x) \frac{d}{dx} f(x) \\ \\ \bigstar \bf\frac{d}{dx} log(f(x)) = \frac{1}{f(x)} . \frac{d}{dx} f(x)\end{gathered}

★

dx

d

f(x)

2

=2f(x)

dx

d

f(x)

★

dx

d

log(f(x))=

f(x)

1

.

dx

d

f(x)

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▪Solution :-

\bf y = [log \{log(logx) \}] {}^{2}y =[log{log(logx)}]

2

Differentiating both side w.r.t x

\begin{gathered} \bf\frac{dy}{dx} = \frac{d}{dx} [log \{log(logx) \}] {}^{2} \\ \\ = \sf2[log \{log(logx) \}] .\frac{d}{dx} [log \{log(logx) \}] \\ \\ = \sf 2[log \{log(logx) \}] \times \frac{1}{ \{log(log \: x) \} } \\ \times \sf\frac{ d}{dx} \{log{(log \: x)} \} \\ \\ = \sf 2[log \{log(logx) \}] \times \frac{1}{ \{log(log \: x) \} } \\ \times \sf\frac{1}{log \: x} \times \frac{d}{dx}log \: x \\ \\ = \sf 2[log \{log(logx) \}] \times \frac{1}{ \{log(log \: x) \} } \\ \times \sf\frac{1}{log \: x} \times \frac{1}{x}\\\\ \colorbox{lime}{ \underline{\boxed{\bf \frac{dy}{dx}=\frac{2[log \{log(logx)\}]}{x log \: x.\{log(log \: x)\} }}}} \end{gathered}

dx

dy

=

dx

d

[log{log(logx)}]

2

=2[log{log(logx)}].

dx

d

[log{log(logx)}]

=2[log{log(logx)}] ×

{log(logx)}

1

×

dx

d

{log(logx)}

=2[log{log(logx)}] ×

{log(logx)}

1

×

logx

1

×

dx

d

logx

=2[log{log(logx)}] ×

{log(logx)}

1

×

logx

1

×

x

1

dx

dy

=

xlogx.{log(logx)}

2[log{log(logx)}]

\begin{gathered} \Large \color{purple}\mathfrak{ \text{W}hich \: \: is \: \: the \: \: required }\\ \huge \color{navy} \mathfrak{ \text{ A}nswer.}\end{gathered}

Which is the required

Answer.

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Answer 2

Answer:

56

Step-by-step explanation:

The given number is less than the number obtained by interchanging the digits by 9. Therefore, (10x+y)−(10y+x)=9. Putting the value of x in x+y=11, we get y=11−x=11−6=5. The number was 56.