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Find the derivative of

y=\sqrt{11 x^{3}-7 x^{2}-\frac{3}{x}}

. A $y^{\prime}=\...

Feb 15, 2024

Find the derivative of

y=\sqrt{11 x^{3}-7 x^{2}-\frac{3}{x}}

. A

y^{\prime}=\frac{33 x^{2}-14 x+\frac{3}{x^{2}}}{2 \sqrt{11 x^{3}-7 x^{2}-\frac{3}{x}}}

y^{\prime}=\frac{11 x^{4}-7 x^{3}-3}{2 \sqrt{33 x^{4}-14 x^{3}+3}}

y^{\prime}=\frac{1}{2} \sqrt{33 x^{2}-14 x-3}

y^{\prime}=\frac{1}{2} \sqrt{33 x^{2}-14 x+\frac{3}{x^{2}}}

\mathrm{E} \circ y^{\prime}=\left(\frac{33}{2} x^{2}-7 x+\frac{3}{2 x^{2}}\right) \sqrt{11 x^{3}-7 x^{2}-\frac{3}{x}}

Generated Graph

Solution by Steps

step 2

The derivative of the inside function

11x^3 - 7x^2 - \frac{3}{x}

33x^2 - 14x + \frac{3}{x^2}

step 3

The derivative of the outside function

\sqrt{u}

with respect to

u

\frac{1}{2\sqrt{u}}

step 4

Applying the chain rule, we multiply the derivative of the inside by the derivative of the outside function

step 5

The final derivative is

y' = \frac{33x^2 - 14x + \frac{3}{x^2}}{2\sqrt{11x^3 - 7x^2 - \frac{3}{x}}}

Key Concept

Chain Rule

Explanation

To differentiate a composite function, apply the chain rule by taking the derivative of the outer function and multiplying it by the derivative of the inner function.