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Worked solution

Differentiate a square root through its inner function

Differentiate 3x−2\sqrt{3x-2} with respect to xx.

CJC · O Level revision · Example 27

Answer

32(3x−2)−1/2=323x−2\dfrac32(3x-2)^{-1/2}=\dfrac3{2\sqrt{3x-2}}, for x>2/3x>2/3.

Compare your reasoning

Use a half power and multiply by the inner derivative

ddx3x−2=ddx(3x−2)1/2=12(3x−2)−1/2ddx(3x−2)=12(3x−2)−1/2⋅3=32(3x−2)−1/2.\begin{aligned}\frac d{dx}\sqrt{3x-2}&=\frac d{dx}(3x-2)^{1/2}\\&=\frac12(3x-2)^{-1/2}\frac d{dx}(3x-2)\\&=\frac12(3x-2)^{-1/2}\cdot3\\&=\frac32(3x-2)^{-1/2}.\end{aligned}

The real function exists for x≥2/3x\ge2/3, but the derivative formula requires x>2/3x>2/3; it has no finite derivative at the endpoint. This distinction is independently supplied.

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