Porisma

Worked solution

Rewrite reciprocal and radical terms as powers

Differentiate 1x3−x5+x3\dfrac1{x^3}-x^5+\sqrt{x^3} with respect to xx.

CJC · O Level revision · Example 24

Answer

−3x−4−5x4+32x1/2-3x^{-4}-5x^4+\dfrac32x^{1/2}, for x>0x>0.

Compare your reasoning

Keep the radical over x cubed

ddx(1x3−x5+x3)=ddx(x−3−x5+x3/2)\begin{aligned}&\frac{d}{dx}\left(\frac1{x^3}-x^5+\sqrt{x^3}\right)\\ & =\frac{d}{dx}\left(x^{-3}-x^5+x^{3/2}\right)\end{aligned}

The printed last term is x3\sqrt{x^3}, which is x3/2x^{3/2} on the real domain. Together with the reciprocal term this requires x>0; that domain qualification is independent.

Differentiate each power

−3x−4−5x4+32x1/2-3x^{-4}-5x^4+\frac32x^{1/2}

Use the constant-multiple and sum/difference rules, preserving each coefficient and exponent.

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