What is the derivative of 1/(\sqrt[3]{(8-x^3)^8)} ?

The derivative of 1/(\sqrt[3]{(8-x^3)^8)} is (8x^2(-x^3+8)^7)/(((8-x^3)^8)^{4/3)}

What is the first derivative of 1/(\sqrt[3]{(8-x^3)^8)} ?

The first derivative of 1/(\sqrt[3]{(8-x^3)^8)} is (8x^2(-x^3+8)^7)/(((8-x^3)^8)^{4/3)}

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derivative of

\frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}

Solution steps

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \frac{d}{d x} (((8 - x^{3})^{8})^{- \frac{1}{3}})

Apply the chain rule:

- \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} \frac{d}{d x} ((8 - x^{3})^{8})

= - \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} \frac{d}{d x} ((8 - x^{3})^{8})

\frac{d}{d x} ((8 - x^{3})^{8}) = - 24 x^{2} (8 - x^{3})^{7}

= - \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} (- 24 x^{2} (8 - x^{3})^{7})

Simplify

- \frac{1}{3 ( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}} (- 24 x^{2} (8 - x^{3})^{7}) : \frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}

= \frac{8 x ^{2} ( - x ^{3} + 8 ) ^{7}}{( ( 8 - x ^{3} ) ^{8} ) ^{\frac{4}{3}}}

What is the derivative of 1/(\sqrt[3]{(8-x^3)^8)} ?
The derivative of 1/(\sqrt[3]{(8-x^3)^8)} is (8x^2(-x^3+8)^7)/(((8-x^3)^8)^{4/3)}
What is the first derivative of 1/(\sqrt[3]{(8-x^3)^8)} ?
The first derivative of 1/(\sqrt[3]{(8-x^3)^8)} is (8x^2(-x^3+8)^7)/(((8-x^3)^8)^{4/3)}