Find the derivative of [3]{3x³+1} with respect to x.

Correct Answer is : 3x2(3x3+1)23{3x²}{[3]{(3x³+1)²}}3(3x3+1)23x2

Correct Answer is : 3x2(3x3+1)23{3x²}{[3]{(3x³+1)²}}3(3x3+1)23x2

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WAEC Further Mathematics 2014 Paper

Question : 13

Total: 40

\sqrt[3]{3x^{3}+1}

with respect to x.

Solution:

Concept:

Use the chain rule to differentiate a composite power function.

Explanation:

Let

y = \sqrt[3]{3x^{3}+1} = (3x^{3}+1)^{\frac{1}{3}}

Set

u = 3x^{3}+1

, so

y = u^{\frac{1}{3}}

Differentiate

y

with respect to

u

\frac{dy}{du} = \frac{1}{3}u^{-\frac{2}{3}}

Differentiate

u

with respect to

x

\frac{du}{dx} = 9x^{2}

By the chain rule,

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}

\frac{dy}{dx} = \frac{1}{3}(3x^{3}+1)^{-\frac{2}{3}} \cdot 9x^{2}

Simplify:

\frac{dy}{dx} = \frac{9x^{2}}{3(3x^{3}+1)^{\frac{2}{3}}} = \frac{3x^{2}}{(3x^{3}+1)^{\frac{2}{3}}}

Rewrite as

\frac{3x^{2}}{\sqrt[3]{(3x^{3}+1)^{2}}}

Answer:

\frac{dy}{dx} = \frac{3x^{2}}{\sqrt[3]{(3x^{3}+1)^{2}}}

, which is Option B.

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