What is the derivative of #3*(sqrt x) - (sqrtx^3)#?

Answer 1

#d/(d x)(3*sqrt x-sqrt (x^3))=3/2*((sqrtx-sqrt2))/x#

#d/(d x)(3*sqrt x-sqrt (x^3))=?# #(3*1)/(2sqrt x)-(3x^2)/(2sqrt(x^3))# #(3*1)/(2sqrt x color(green)(sqrt (x^3)))-(3x^2)/(2sqrt(x^3)color(green)(sqrt2))# #(3sqrt (x^3)-3x^2sqrt2)/(2sqrt(x^4))# #(3sqrt (x^3)-3x^2sqrt2)/(2x^2)# #(3cancel(x))/(2)((sqrtx-xsqrt2)/cancel((x^2)))# #d/(d x)(3*sqrt x-sqrt (x^3))=3/2*((sqrtx-sqrt2))/x#

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Answer 2

William Abdalla

The derivative of (3\sqrt{x} - \sqrt{x^3}) is (\frac{3}{2\sqrt{x}} - \frac{3x}{2\sqrt{x^3}}).

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Answer 3

Christopher Agresta

The derivative of (3\sqrt{x} - \sqrt{x^3}) with respect to (x) is:

[ \frac{d}{dx} \left( 3\sqrt{x} - \sqrt{x^3} \right) = \frac{3}{2\sqrt{x}} - \frac{3x^{3/2}}{2\sqrt{x}} ]

Simplifying:

[ \frac{d}{dx} \left( 3\sqrt{x} - \sqrt{x^3} \right) = \frac{3}{2\sqrt{x}} - \frac{3x}{2\sqrt{x}} = \frac{3 - 3x}{2\sqrt{x}} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.