d[Cos-1((4x3/27)-x)]/dx=? a)3/(9-x2)1/2 b)Negative of the above c)1/(3-x2)1/2 d)Negative of the above option Plz explain..

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Differential Calculus> Differentiation...

d[Cos^-1((4x³/27)-x)]/dx=?

a)3/(9-x²)^1/2

b)Negative of the above

c)1/(3-x²)^1/2

d)Negative of the above option

Plz explain..

Chilukuri Sai Kartik , 14 Years ago

Grade 12

3 Answers

Arun Kumar

Last Activity: 10 Years ago

Hello

$\\\frac{d}{dx}\left(\arccos \left(\frac{4x^3}{27}-x\right)\right) \\=\left(-\frac{1}{\sqrt{1-u^2}}\right)\left(\frac{4x^2}{9}-1\right) \\u=\frac{4x^3}{27}-x \\=-\frac{3\left(4x^2-9\right)}{\sqrt{-\left(9-4x^2\right)^2\left(x^2-9\right)}}$

Thanks & Regards
Arun Kumar
Btech IIT delhi
Askiitians Faculty

Ruthwik

Last Activity: 5 Years ago

4x^3-27x/27=4(x/3)^3-x____(1)

Let x/3=cos theta

Theta = cos^-1x/3

Multiply and divide (1) with 3

4(x/3)^3-3x/3

4cos^3theta-3costheta=cos3theta

Substitute the value of cos3theta in the given question

I.e, cos^-1(4×^3-27×/27)

We get cos^-1(cos3theta)

I.e, 3theta=y

Dy/dx= d/dx(3theta)

3d/dxtheta

Theta=cos^-1 ×/3

3(-1/(1-×^2/9)^1/2

3{-1/(9-×^2/9)^1/2}

Sandeep

Last Activity: 4 Years ago

By substitution method

we can convert (4x³/27-x) of type (4cos³y-3cosy) which is equal to cos3y

so put x/3=cosy ----1

Then it would be

let z= cos^-1[4x³/27-x]

from eq--1

z=cos^-1[4cos³y-cosy]

z=cos^-1[cos3y]

z=3y

as x/3=cosy

y=cos^-1(x/3)

z=3cos^-1(x/3)

Now

dz/dx =3*dcos^-1(x/3)/dx

dcos^-1x/dx= -1/(1-x²)^{½. ----formula}

^{let t=x/3}

^{By chain rule}

dz/dx=3[dcos^-1t/dt*dt/dx]

dz/dx= 3[-1/(1+(x/3)^1/2 *d(x/3)/dx]

dz/dx= 3* [-3/(9+x²)*1/3]

dz/dx= -3/(9+x²) ----(answer)

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Differential Calculus> Differentiation...

d[Cos-1((4x3/27)-x)]/dx=? a)3/(9-x2)1/2 b)Negative of the above c)1/(3-x2)1/2 d)Negative of the above option Plz explain..

Chilukuri Sai Kartik , 14 Years ago

Arun Kumar

Ruthwik

Sandeep

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d[Cos^-1((4x³/27)-x)]/dx=?

a)3/(9-x²)^1/2

b)Negative of the above

c)1/(3-x²)^1/2

d)Negative of the above option

Plz explain..

Find dy/DX when y=x^sinx+sin-1√x.

Please give me the solution of this problem.

Example 20. (i) In a AABC, find the maxima and minima of u = sin A sin B sin C
where A + B + C = T
(ii) In a plane triangle ABC, find the maximum value of cosA cosB cosC.