Find dy/dx, if y = (sin x) ^x +(cos x) ^tan x​

Question

Find dy/dx, if y = (sin x) ^x +(cos x) ^tan x​

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Evelyn 4 weeks 2021-09-22T00:03:45+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-09-22T00:05:03+00:00

    Answer:

    \begin{gathered}{\frak{Given}} \begin{cases} & \textsf{Radius of cone = 3.5 cm } \\ & \textsf{Height of cone = 15.5 cm} \end{cases}\end{gathered}

    Given{

    Radius of cone = 3.5 cm

    Height of cone = 15.5 cm

    We have to find, Total surface area of toy.

    ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

    ☯ Toy is hemispherical at bottom and conical at top.

    ⠀⠀⠀⠀⠀⠀⠀

    \setlength{\unitlength}{1cm}\begin{picture}(6, 4)\linethickness{0.26mm}\qbezier(5.8,2.0)(5.8,2.3728)(4.9799,2.6364)\qbezier(4.9799,2.6364)(4.1598,2.9)(3.0,2.9)\qbezier(3.0,2.9)(1.8402,2.9)(1.0201,2.6364)\qbezier(1.0201,2.6364)(0.2,2.3728)(0.2,2.0)\qbezier(0.2,2.0)(0.2,1.6272)(1.0201,1.3636)\qbezier(1.0201,1.3636)(1.8402,1.1)(3.0,1.1)\qbezier(3.0,1.1)(4.1598,1.1)(4.9799,1.3636)\qbezier(4.9799,1.3636)(5.8,1.6272)(5.8,2.0)\put(0.2,2){\line(1,0){5.6}}\put(3,2){\line(0,2){4.5}}\put(1.5,1.7){\sf{3.5 cm}}\qbezier(.2,2.05)(.7,3)(3,6.5)\qbezier(5.8,2.05)(5.3,3)(3,6.5)\put(2,4){\sf 12 cm}\put(3,2.02){\circle*{0.15}}\put(2.7,2){\dashbox{0.01}(.3,.3)}\qbezier(0.2,2)(2.9,-2)(5.8,2)\end{picture}

    ⠀⠀⠀⠀⠀⠀⠀

    Therefore

    TSA of toy = CSA of hemisphere + CSA of cone

    ⠀⠀⠀⠀⠀⠀⠀

    Slant height of cone, l = \sf \sqrt{h^2 + r^2}

    h

    2

    +r

    2

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf l = \sqrt{3.5^2 + 12^2}:⟹l=

    3.5

    2

    +12

    2

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf l = \sqrt{12.25 + 144}:⟹l=

    12.25+144

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf l = \sqrt{156.25}:⟹l=

    156.25

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies{\underline{\boxed{\sf{\pink{l = 12.5\;cm}}}}}\;\bigstar:⟹

    l=12.5cm

    ━━━━━━━━━━━━━━━━━━━━━

    Therefore,

    \;\;\;\;\;\;\;\star\; 2 \pi r^2 + \pi rl⋆2πr

    2

    +πrl

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 2 \pi (3.5)^2 + \pi (3.5)(12.5):⟹2π(3.5)

    2

    +π(3.5)(12.5)

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 24.5 \pi + 43.75 \pi:⟹24.5π+43.75π

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 68.25 \pi:⟹68.25π

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies\sf 68.25 \times \dfrac{22}{7}:⟹68.25×

    7

    22

    ⠀⠀⠀⠀⠀⠀⠀

    :\implies{\underline{\boxed{\sf{\purple{l = 214 5\;cm}}}}}\;\bigstar:⟹

    l=2145cm

    ⠀⠀⠀⠀⠀⠀⠀

    \therefore∴ Hence, Total surface area of toy is 214.5 cm².

    0
    2021-09-22T00:05:32+00:00

    Answer:

    \bold{\purple{\fbox{\red{Answer}}}}

    y =  {sinx}^{x}  +  {cosx}^{tanx}.........(1)

     \:  \:  \: to \: find =  \frac{dy}{dx}

    y =  {sinx}^{x}  +  {cosx}^{tanx}

    by \: getting \: log \: both \: side

    logy = log {sin}^{x}  + log {cosx}^{tanx}

    logy = xlogsinx + tanxlogcosx

    by \: product\: rule

     \frac{1}{y}  \times  \frac{dy}{dx} =  logsinx \times  \frac{1}{sinx} \times cosx  +  {secx}^{2}  \times  logcosx \:  \times  \frac{1}{cosx}  \times  - sinx

    just \: simplify

     \frac{dy}{dx}  = y(logsinx \times  \frac{cosx}{sinx}  +  {secx}^{2}  \times logcosx \times  - \frac{sinx}{cosx}

     \frac{dy}{dx }  = y(logsinx \times cotx +  {secx}^{2}  \times logcosx -  tanx)

     \frac{dy}{dx}  = y(logsinx \: cotx -  {secx}^{2}logcosx \: tanx)

    from \: equation \: (1) \: answer \: will \: be

     \frac{dy}{dx}  =  {sin}^{x}  +  {cosx}^{tanx} (logsinxcotx -  {secx}^{2}logcosxtanx)

    ☞ correct\large\blue{\rm\:Answer}

     \frac{dy}{dx  }  =  {sin}^{x}  +  {cosx}^{tanx} (logsinxcotx -  {secx}^{2} logcosxtanx)

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