Derivative of 1/cube root x

The derivative of 1/(cube root of x) is equal to $-\dfrac{1}{3\sqrt[3]{x^4}}$. Note that 1/(cube root of x) can be written mathematically as $\frac{1}{\sqrt[3]{x}}$. In this post, we will learn how to differentiate 1/(cube root of x).

Derivative of 1/cube root x by Power Rule

To find the derivative of 1/cube root of x, we will follow the below steps.

Step 1: At first, we will write $\dfrac{1}{\sqrt[3]{x}}$ as a power of x using the rule of indices. See that

$\dfrac{1}{\sqrt[3]{x}}$ = $\dfrac{1}{x^{1/3}}$ as cube root is the same as power 1/3.

= $x^{-1/3}$ as we know that 1/a^m=a^-m

Step 2: By step 1, we get that

$\dfrac{d}{dx}(\dfrac{1}{\sqrt[3]{x}})$ = $\dfrac{d}{dx}(x^{-1/3})$

= $-\dfrac{1}{3} x^{-\frac{1}{3}-1}$ by the power rule of derivatives d/dx(xⁿ)=nx^n-1

Step 3: Simplify the above expression.

$\dfrac{d}{dx}(\dfrac{1}{\sqrt[3]{x}})$ = $-\dfrac{1}{3} x^{-\frac{4}{3}}$

= $-\dfrac{1}{3x^{\frac{4}{3}}}$

= $-\dfrac{1}{3\sqrt[3]{x^4}}$

So the derivative of 1 divided by cube root x is equal to $-\dfrac{1}{3\sqrt[3]{x^4}}$. This is proved above using the power rule of derivatives and the rule of indices.

Question: Find the derivative of $\dfrac{1}{\sqrt[3]{x}}$ at x=1.

By above, we obtain that

$\Big[ \dfrac{d}{dx}(\dfrac{1}{\sqrt[3]{x}}) \Big]_{x=1}$ = $\Big[ -\dfrac{1}{3\sqrt[3]{x^4}} \Big]_{x=1}$

= $-\dfrac{1}{3\sqrt[3]{1^4}}$

= $-\dfrac{1}{3}$

So the derivative of 1/cube root x at the point x=1 is equal to -1/3.

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