Problem

If $y=2 u^{2}-u+5$, and $u=1-x^{2}$, find $d y / d x$.

Answer

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Answer

\(\boxed{\frac{dy}{dx} = 8x^3 - 6x}\)

Steps

Step 1 :\(\frac{dy}{du} = \frac{d}{du}(2u^2 - u + 5)\)

Step 2 :\(\frac{dy}{du} = 4u - 1\)

Step 3 :\(\frac{du}{dx} = \frac{d}{dx}(1 - x^2)\)

Step 4 :\(\frac{du}{dx} = -2x\)

Step 5 :\(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Step 6 :\(\frac{dy}{dx} = (4u - 1) \cdot -2x\)

Step 7 :\(\frac{dy}{dx} = -8ux + 2x\)

Step 8 :Substitute \(u = 1 - x^2\) into the equation

Step 9 :\(\frac{dy}{dx} = -8x(1 - x^2) + 2x\)

Step 10 :\(\frac{dy}{dx} = -8x + 8x^3 + 2x\)

Step 11 :\(\frac{dy}{dx} = 8x^3 - 6x\)

Step 12 :\(\boxed{\frac{dy}{dx} = 8x^3 - 6x}\)

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