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(a) y=2(4x+1)³, (y/2)=(4x+1)³, (y/2)^⅓=4x+1, (y/2)^⅓-1=4x, x=((y/2)^⅓-1)/4, is the inverse function.

Differentiating wrt x: 1=[(1/24)(y/2)^-⅔](dy/dx). dy/dx=24(y/2)^⅔.

[Since y=2(4x+1)³, y/2=(4x+1)³ and (y/2)^⅔=(4x+1)², dy/dx=24(4x+1)², which is what we get by direct differentiation.]

(b) Did you mean y=(4-3x)^(3/2)? If so:

y^⅔=4-3x, 3x=4-y^⅔, x=⅓(4-y^⅔) is the inverse function.

Differentiating: 1=[-(2/9)y^-⅓](dy/dx), dy/dx=-(9/2)y^⅓.

[Directly, dy/dx=-(9/2)(4-3x)^½. y^⅓=(4-3x)^½, so dy/dx=-(9/2)(4-3x)^½.]

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by Top Rated User (762k points)
thank you so much sir

You’re very welcome!

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