Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .
A) $80 x ^ { 3 }$
B) $- \frac { 80 } { 11 } x ^ { 3 }$
C) $60 x ^ { 2 } + 11$
D) $60 x ^ { 2 } - 100$

Question

Quizplus · Accepted Answer

To find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$, we first differentiate both sides with respect to $x$ to get the first derivative, $y'$. Differentiating $4x^5$ gives $20x^4$, and differentiating $11y$ using the chain rule gives $11y'$. Setting the equation equal to 0 gives $20x^4 + 11y' = 0$, so $y' = -\frac{20}{11}x^4$. Differentiating $y'$ again with respect to $x$ to find $y''$ gives $\frac{d}{dx}(-\frac{20}{11}x^4) = -\frac{80}{11}x^3$, which is option B.

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Use Implicit Differentiation to Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$