What is the derivative of \sqrt[3]{a+bx^3} ?

The derivative of \sqrt[3]{a+bx^3} is (bx^2)/((a+bx^3)^{2/3)}

What is the first derivative of \sqrt[3]{a+bx^3} ?

The first derivative of \sqrt[3]{a+bx^3} is (bx^2)/((a+bx^3)^{2/3)}

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\frac{b x ^{2}}{( a + b x ^{3} ) ^{\frac{2}{3}}}

Solution steps

Apply the chain rule:

\frac{1}{3 ( a + b x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (a + b x^{3})

= \frac{1}{3 ( a + b x ^{3} ) ^{\frac{2}{3}}} \frac{d}{d x} (a + b x^{3})

\frac{d}{d x} (a + b x^{3}) = 3 b x^{2}

= \frac{1}{3 ( a + b x ^{3} ) ^{\frac{2}{3}}} \cdot 3 b x^{2}

Simplify

\frac{1}{3 ( a + b x ^{3} ) ^{\frac{2}{3}}} \cdot 3 b x^{2} : \frac{b x ^{2}}{( a + b x ^{3} ) ^{\frac{2}{3}}}

= \frac{b x ^{2}}{( a + b x ^{3} ) ^{\frac{2}{3}}}

f (x) = 5 x

\int_{1}^{4} x (1 + x) d x

sin^{4} (x)

\int \frac{2 u ^{2} - 1}{u ^{3} - u} d u

What is the derivative of \sqrt[3]{a+bx^3} ?
The derivative of \sqrt[3]{a+bx^3} is (bx^2)/((a+bx^3)^{2/3)}
What is the first derivative of \sqrt[3]{a+bx^3} ?
The first derivative of \sqrt[3]{a+bx^3} is (bx^2)/((a+bx^3)^{2/3)}